6434:
below. This increase in spatial order has certain advantages over 2nd order schemes for smooth solutions, however, for shocks it is more dissipative - compare diagram opposite with above solution obtained using the KT algorithm with linear extrapolation and
Superbee limiter. This simulation was carried out on a mesh of 200 cells using the same KT algorithm but with parabolic reconstruction. Time integration was by RK-4, and the alternative form of van Albada limiter,
803:
8277:
7404:
5979:
76:
4810:
8223:
3317:
7913:
7660:
7129:
1186:
5146:
2227:
2481:
6742:
4817:
It is possible to extend the idea of linear-extrapolation to higher order reconstruction, and an example is shown in the diagram opposite. However, for this case the left and right states are estimated by interpolation of a second-order, upwind biased, difference equation. This results in a parabolic
3958:
and used RK-4 for time integration. This simulation result contrasts extremely well against the above first-order upwind and second-order central difference results shown above. This scheme also provides good results when applied to sets of equations - see results below for this scheme applied to the
8235:
problem (Sod, 1978) using the above high resolution
Kurganov and Tadmor Central Scheme (KT) with Linear Extrapolation and Ospre limiter. This illustrates clearly demonstrates the effectiveness of the MUSCL approach to solving the Euler equations. The simulation was carried out on a mesh of 200 cells
626:
8289:
problem (Sod, 1978) using the above high resolution
Kurganov and Tadmor Central Scheme (KT) but with parabolic reconstruction and van Albada limiter. This again illustrates the effectiveness of the MUSCL approach to solving the Euler equations. The simulation was carried out on a mesh of 200 cells
406:
This basic scheme is not able to handle shocks or sharp discontinuities as they tend to become smeared. An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation with a step wave propagating to the right. The simulation was carried out with a mesh of 200
6433:
Parabolic reconstruction is straight forward to implement and can be used with the
Kurganov and Tadmor scheme in lieu of the linear extrapolation shown above. This has the effect of raising the spatial solution of the KT scheme to 3rd order. It performs well when solving the Euler equations, see
7668:
7415:
1964:
56:
by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited, reconstructed left and right states are obtained and used to calculate fluxes at the cell boundaries (edges). These fluxes can, in turn, be used as input to a
6391:
7399:{\displaystyle \rho _{i+{\frac {1}{2}}}^{*}=\rho _{i+{\frac {1}{2}}}^{*}\left(\rho _{i+{\frac {1}{2}}}^{L},\rho _{i+{\frac {1}{2}}}^{R}\right),\quad \rho _{i-{\frac {1}{2}}}^{*}=\rho _{i-{\frac {1}{2}}}^{*}\left(\rho _{i-{\frac {1}{2}}}^{L},\rho _{i-{\frac {1}{2}}}^{R}\right),}
6216:
4921:
8280:
High resolution simulation of Euler equations based on G A Sod's 'Shock Tube' problem - SI units. Shows the analytical solutions along with simulated (3rd order) solutions based upon the
Kurganov and Tadmor Central Scheme with parabolic reconstruction and van Albada
4294:
6595:
8226:
High resolution simulation of Euler equations based on G A Sod's 'Shock Tube' problem. Shows the analytical solutions along with simulated (2nd order) solutions based upon the
Kuganov and Tadmor Central Scheme with Linear Extrapolation and Ospre
8163:
3311:
3051:
4793:. A later paper (Kurganov and Levy, 2000) demonstrates that it can also form the basis of a third order scheme. A 1D advective example and an Euler equation example of their scheme, using parabolic reconstruction (3rd order), are shown in the
3819:
1975:
4782:
versions) and its derivation can be found in the original paper (Kurganov and Tadmor, 2000), along with a number of 1D and 2D examples. Additional information is also available in the earlier related paper by
Nessyahu and Tadmor (1990).
5773:
5561:
2235:
1526:
1370:
797:
401:
5973:
5349:
6606:
428:
7908:{\displaystyle \rho _{i-{\frac {1}{2}}}^{L}=\rho _{i-1}+0.5\phi \left(r_{i-1}\right)\left(\rho _{i}-\rho _{i-1}\right),\quad \rho _{i-{\frac {1}{2}}}^{R}=\rho _{i}-0.5\phi \left(r_{i}\right)\left(\rho _{i+1}-\rho _{i}\right).}
7655:{\displaystyle \rho _{i+{\frac {1}{2}}}^{L}=\rho _{i}+0.5\phi \left(r_{i}\right)\left(\rho _{i}-\rho _{i-1}\right),\quad \rho _{i+{\frac {1}{2}}}^{R}=\rho _{i+1}-0.5\phi \left(r_{i+1}\right)\left(\rho _{i+1}-\rho _{i}\right),}
1682:
3887:
8171:
The above illustrates the basic idea of the MUSCL scheme. However, for a practical solution to the Euler equations, a suitable scheme (such as the above KT scheme), also has to be chosen in order to define the function
2629:
is a limiter function that limits the slope of the piecewise approximations to ensure the solution is TVD, thereby avoiding the spurious oscillations that would otherwise occur around discontinuities or shocks - see
1131:(TVD) scheme and introduces spurious oscillations into the solution where discontinuities or shocks are present. An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation
4056:
6224:
3741:
2586:
6061:
6028:, with step wave propagating to the right. Shows the analytical solution along with a simulation based upon the Kurganov and Tadmor Central Scheme with parabolic reconstruction and van Albada limiter.
5141:{\displaystyle u_{i+{\frac {1}{2}}}^{*}=f\left(u_{i+{\frac {1}{2}}}^{L},u_{i+{\frac {1}{2}}}^{R}\right),\quad u_{i-{\frac {1}{2}}}^{*}=f\left(u_{i-{\frac {1}{2}}}^{L},u_{i-{\frac {1}{2}}}^{R}\right),}
4527:
4769:
1122:
1032:
8217:
8394:
More information on these and other methods can be found in the references below. An open source implementation of the
Kurganov and Tadmor central scheme can be found in the external links below.
8367:
6507:
6953:
129:
We will consider the fundamentals of the MUSCL scheme by considering the following simple first-order, scalar, 1D system, which is assumed to have a wave propagating in the positive direction,
4067:
7040:
3467:
8004:
Having obtained the limited extrapolated states, we then proceed to construct the edge fluxes using these values. With the edge fluxes known, we can now construct the semi-discrete scheme,
188:
3415:
2782:
The KT scheme extends the NT scheme and has a smaller amount of numerical viscosity than the original NT scheme. It also has the added advantage that it can be implemented as either a
4909:
4864:
240:
The basic scheme of
Godunov uses piecewise constant approximations for each cell, and results in a first-order upwind discretisation of the above problem with cell centres indexed as
2627:
3959:
Euler equations. However, care has to be taken in choosing an appropriate limiter because, for example, the
Superbee limiter can cause unrealistic sharpening for some smooth waves.
6428:
8485:
31:
that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for
8646:
6527:
1575:
3952:
1176:
852:, with step wave propagating to the right. Shows the analytical solution along with a simulation based upon a second order, central difference spatial discretization scheme.
8014:
6026:
3364:
3059:
2799:
2733:
1670:
1624:
850:
123:
3503:
6796:
6770:
4911:
again represent scheme dependent functions (of the limited reconstructed cell edge variables). But for this case they are based upon parabolically reconstructed states,
3366:, with step wave propagating to the right. Shows the analytical solution along with a simulation based upon the Kurganov and Tadmor central scheme with SuperBee limiter.
7066:
6053:
2222:{\displaystyle u_{i+1/2}^{L}=u_{i}+0.5\phi \left(r_{i}\right)\left(u_{i+1}-u_{i}\right),u_{i+1/2}^{R}=u_{i+1}-0.5\phi \left(r_{i+1}\right)\left(u_{i+2}-u_{i+1}\right),}
936:
895:
8472:
7123:
We can now proceed, as shown above in the simple 1D example, by obtaining the left and right extrapolated states for each state variable. Thus, for density we obtain
3749:
2658:
2476:{\displaystyle u_{i-1/2}^{L}=u_{i-1}+0.5\phi \left(r_{i-1}\right)\left(u_{i}-u_{i-1}\right),u_{i-1/2}^{R}=u_{i}-0.5\phi \left(r_{i}\right)\left(u_{i+1}-u_{i}\right),}
7939:
2775:
problems, and can be viewed as a Rusanov flux (also called the local Lax-Friedrichs flux) supplemented with high order reconstructions. The algorithm is based upon
7118:
6832:
5569:
5357:
8491:
6979:
2684:
1381:
1203:
637:
266:
6737:{\displaystyle \mathbf {U} ={\begin{pmatrix}\rho \\\rho u\\E\end{pmatrix}}\qquad \mathbf {F} ={\begin{pmatrix}\rho u\\p+\rho u^{2}\\u(E+p)\end{pmatrix}},\qquad }
3962:
The scheme can readily include diffusion terms, if they are present. For example, if the above 1D scalar problem is extended to include a diffusion term, we get
7999:
7979:
7959:
6892:
6872:
6852:
258:
231:
211:
5781:
5157:
125:, with step wave propagating to the right. Shows the analytical solution along with a simulation based upon a first order upwind spatial discretization scheme.
621:{\displaystyle u\left(x\right)=u_{i}+{\frac {\left(x-x_{i}\right)}{\left(x_{i+1}-x_{i}\right)}}\left(u_{i+1}-u_{i}\right)\qquad \forall x\in (x_{i},x_{i+1}].}
8529:
414:
To provide higher resolution of discontinuities, Godunov's scheme can be extended to use piecewise linear approximations of each cell, which results in a
3892:
1959:{\displaystyle u_{i+1/2}^{*}=u_{i+1/2}^{*}\left(u_{i+1/2}^{L},u_{i+1/2}^{R}\right),u_{i-1/2}^{*}=u_{i-1/2}^{*}\left(u_{i-1/2}^{L},u_{i-1/2}^{R}\right),}
9116:
8737:
3954:, with a step wave propagating to the right. The simulation was carried out on a mesh of 200 cells, using the Kurganov and Tadmor central scheme with
2779:
with comparable performance to Riemann type solvers when used to obtain solutions for PDE's describing systems that exhibit high-gradient phenomena.
8687:
3828:
8663:
8637:
2686:. Thus, the accuracy of a TVD discretization degrades to first order at local extrema, but tends to second order over smooth parts of the domain.
8369:, was used to avoid spurious oscillations. Time integration was performed by a 4th order SHK integrator. The same initial conditions were used.
3905:
An example of the effectiveness of using a high resolution scheme is shown in the diagram opposite, which illustrates the 1D advective equation
8476:
Kurganov, Alexander and Doron Levy (2000), A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations,
8455:
9121:
8961:
8731:
9031:
8888:
8743:
8372:
Various other high resolution schemes have been developed that solve the Euler equations with good accuracy. Examples of such schemes are,
3968:
6517:
For simplicity we consider the 1D case without heat transfer and without body force. Therefore, in conservation vector form, the general
6386:{\displaystyle \delta u_{i+{\frac {3}{2}}}=\left(u_{i+2}-u_{i+1}\right),\quad \delta u_{i-{\frac {3}{2}}}=\left(u_{i-1}-u_{i-2}\right),}
6211:{\displaystyle \delta u_{i+{\frac {1}{2}}}=\left(u_{i+1}-u_{i}\right),\quad \delta u_{i-{\frac {1}{2}}}=\left(u_{i}-u_{i-1}\right),}
8240:. Time integration was performed by a 4th order SHK (equivalent performance to RK-4) integrator. The following initial conditions (
3511:
2492:
8956:
8939:
6518:
63:, following which the solutions are averaged and used to advance the solution in time. Alternatively, the fluxes can be used in
9090:
8876:
4305:
4535:
1038:
948:
8857:
8846:
8823:
4821:
We follow the approach of Kermani (Kermani, et al., 2003), and present a third-order upwind biased scheme, where the symbols
8175:
8297:
6437:
8829:
8450:
Kermani, M. J., Gerber, A. G., and Stockie, J. M. (2003), Thermodynamically Based Moisture Prediction Using Roe’s Scheme,
4289:{\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}=-{\frac {1}{\Delta x_{i}}}\left+{\frac {1}{\Delta x_{i}}}\left.}
20:
8946:
8911:
8505:
van Leer, B. (1979), Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov's Method,
6900:
2767:
that uses MUSCL reconstruction. It is a fully discrete method that is straight forward to implement and can be used on
8951:
8630:
8433:
1128:
45:
9068:
6994:
3420:
135:
9053:
8929:
3902:(though it's worth mentioning that such flux expression does not appear in Lax, 1954 but rather on Rusanov, 1961).
8695:
8677:
8715:
8463:(2000), New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations,
1577:
correspond to a nonlinear combination of first and second-order approximations to the continuous flux function.
9038:
8924:
8654:
3377:
8700:
4869:
4824:
1197:
left and right extrapolated states. This results in the following high resolution, TVD discretisation scheme,
1193:
MUSCL based numerical schemes extend the idea of using a linear piecewise approximation to each cell by using
2594:
9111:
9080:
9058:
9043:
9026:
8934:
8919:
8835:
6399:
9000:
8771:
8623:
8418:
6590:{\displaystyle {\frac {\partial \mathbf {U} }{\partial t}}+{\frac {\partial \mathbf {F} }{\partial x}}=0,}
2764:
9048:
8894:
8810:
8608:– Open source code solving the Euler Equations using the Kurganov and Tadmor central scheme, written in
8615:
8489:
Lax, P. D. (1954). Weak Solutions of Non-linear Hyperbolic Equations and Their Numerical Computation,
8290:
using Matlab code (Wesseling, 2001), adapted to use the KT algorithm with Parabolic Extrapolation and
8158:{\displaystyle {\frac {\mathrm {d} \mathbf {U} _{i}}{\mathrm {d} t}}=-{\frac {1}{\Delta x_{i}}}\left.}
3306:{\displaystyle F_{i+{\frac {1}{2}}}^{*}={\frac {1}{2}}\left\{\left-a_{i+{\frac {1}{2}}}\left\right\}.}
3046:{\displaystyle F_{i-{\frac {1}{2}}}^{*}={\frac {1}{2}}\left\{\left-a_{i-{\frac {1}{2}}}\left\right\}.}
1534:
9085:
8758:
8413:
8403:
3908:
2768:
1179:
1134:
28:
8852:
8766:
5985:
3323:
2692:
1629:
1583:
1127:
Although the above second-order scheme provides greater accuracy for smooth solutions, it is not a
809:
82:
53:
3472:
9075:
9016:
2772:
6777:
6751:
3814:{\displaystyle \rho \left({\frac {\partial F\left(u\left(t\right)\right)}{\partial u}}\right)\ }
7048:
6035:
900:
859:
8705:
8428:
6985:
2776:
802:
8534:
Rusanov, V. V. (1961). Calculation of Intersection of Non-Steady Shock Waves with Obstacles,
5768:{\displaystyle u_{i-{\frac {1}{2}}}^{L}=u_{i-1}+{\frac {\phi \left(r_{i-1}\right)}{4}}\left,}
5556:{\displaystyle u_{i+{\frac {1}{2}}}^{R}=u_{i+1}-{\frac {\phi \left(r_{i+1}\right)}{4}}\left,}
4789:
This scheme was originally presented by Kurganov and Tadmor as a 2nd order scheme based upon
2637:
1182:. The simulation was carried out with a mesh of 200 cells and used RK4 for time integration.
9021:
9011:
8900:
8868:
7921:
1521:{\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}+{\frac {1}{\Delta x_{i}}}\left=0.}
1365:{\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}+{\frac {1}{\Delta x_{i}}}\left=0.}
1178:, with a step wave propagating to the right. This loss of accuracy is to be expected due to
792:{\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}+{\frac {1}{\Delta x_{i}}}\left=0,}
396:{\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}+{\frac {1}{\Delta x_{i}}}\left=0.}
7071:
6817:
408:
9063:
9006:
8995:
8423:
8276:
3822:
6961:
5968:{\displaystyle u_{i-{\frac {1}{2}}}^{R}=u_{i}-{\frac {\phi \left(r_{i}\right)}{4}}\left.}
5344:{\displaystyle u_{i+{\frac {1}{2}}}^{L}=u_{i}+{\frac {\phi \left(r_{i}\right)}{4}}\left,}
2663:
1672:
represent scheme dependent functions (of the limited extrapolated cell edge variables),
8841:
8788:
8438:
7984:
7964:
7944:
6877:
6857:
6837:
2739:(see below), the solution can proceed using standard numerical integration techniques.
243:
216:
196:
59:
5978:
4061:
for which Kurganov and Tadmor propose the following central difference approximation,
9105:
8710:
36:
631:
Thus, evaluating fluxes at the cell edges we get the following semi-discrete scheme
8882:
8799:
8517:
8460:
8408:
8291:
8237:
3955:
2631:
75:
8168:
The solution can now proceed by integration using standard numerical techniques.
8793:
8671:
8520:(1990), Non-oscillatory central differencing for hyperbolic conservation laws,
8286:
8232:
4809:
3882:{\displaystyle {\frac {\partial F\left(u\left(t\right)\right)}{\partial u}}.}
8222:
3316:
2763:, (Nessyahu and Tadmor, 1990). It is a Riemann-solver-free, second-order,
2689:
The algorithm is straight forward to implement. Once a suitable scheme for
8236:
using Matlab code (Wesseling, 2001), adapted to use the KT algorithm and
422:
accurate in space. The piecewise linear approximations are obtained from
8545:
Sod, G. A. (1978), A Numerical Study of a Converging Cylindrical Shock.
8979:
8609:
1185:
8818:
8605:
3417:, is the maximum absolute value of the eigenvalue of the Jacobian of
8454:, Amir Kabir University of Technology, Tehran, Iran, January 27–29.
4051:{\displaystyle u_{t}+F_{x}\left(u\right)=Q_{x}\left(u,u_{x}\right),}
35:(van Leer, 1979), and the term was introduced in a seminal paper by
1189:
An example of MUSCL type left and right state linear-extrapolation.
8275:
8221:
5977:
4808:
3315:
1184:
801:
74:
8973:
8967:
8782:
8387:
1375:
Which, alternatively, can be written in the more succinct form,
234:
8619:
52:
The idea is to replace the piecewise constant approximation of
49:(TVD) scheme where he obtained second order spatial accuracy.
8285:
The diagram opposite shows a 3rd order solution to G A Sod's
8231:
The diagram opposite shows a 2nd order solution to G A Sod's
4818:
reconstruction scheme that is third-order accurate in space.
3736:{\displaystyle a_{i+{\frac {1}{2}}}\left(t\right)=\max \left}
2581:{\displaystyle r_{i}={\frac {u_{i}-u_{i-1}}{u_{i+1}-u_{i}}}.}
938:
are the piecewise approximate values of cell edge variables,
8241:
3895:
related speeds, no characteristic information is required.
4522:{\displaystyle P_{i+{\frac {1}{2}}}={\frac {1}{2}}\left,}
39:(van Leer, 1979). In this paper he constructed the first
8558:
Riemann Solvers and Numerical Methods for Fluid Dynamics
4813:
An example of MUSCL type state parabolic-reconstruction.
4764:{\displaystyle P_{i-{\frac {1}{2}}}={\frac {1}{2}}\left}
1117:{\displaystyle u_{i-1/2}=0.5\left(u_{i-1}+u_{i}\right).}
1027:{\displaystyle u_{i+1/2}=0.5\left(u_{i}+u_{i+1}\right),}
33:
Monotonic Upstream-centered Scheme for Conservation Laws
8212:{\displaystyle \mathbf {F} _{i\pm {\frac {1}{2}}}^{*}}
6782:
6756:
6668:
6623:
16:
Finite volume method in partial differential equations
8362:{\displaystyle \phi _{va}(r)={\frac {2r}{1+r^{2}}}\ }
8300:
8178:
8017:
7987:
7967:
7947:
7924:
7671:
7418:
7132:
7074:
7051:
6997:
6964:
6903:
6880:
6860:
6840:
6820:
6814:. There are thus three equations and four unknowns,
6780:
6754:
6609:
6530:
6502:{\displaystyle \phi _{va}(r)={\frac {2r}{1+r^{2}}}\ }
6440:
6402:
6227:
6064:
6038:
5988:
5784:
5572:
5360:
5160:
4924:
4872:
4827:
4538:
4308:
4070:
3971:
3911:
3898:
The above flux calculation is most frequently called
3831:
3752:
3514:
3475:
3423:
3380:
3326:
3062:
2802:
2695:
2666:
2640:
2597:
2495:
2238:
1978:
1685:
1632:
1586:
1537:
1384:
1206:
1137:
1041:
951:
903:
862:
812:
640:
431:
269:
260:. A semi-discrete scheme can be defined as follows,
246:
219:
199:
138:
85:
8647:
Numerical methods for partial differential equations
8579:
Numerical Computation of Internal and External Flows
8988:
8910:
8867:
8809:
8757:
8724:
8686:
8662:
8653:
2790:scheme. Here we consider the semi-discrete scheme.
67:schemes, which are basically Rusanov-like schemes.
8361:
8211:
8157:
7993:
7973:
7953:
7933:
7907:
7654:
7398:
7112:
7060:
7034:
6973:
6948:{\displaystyle E=\rho e+{\frac {1}{2}}\rho u^{2},}
6947:
6886:
6866:
6846:
6826:
6790:
6764:
6736:
6589:
6501:
6422:
6385:
6210:
6047:
6020:
5967:
5767:
5555:
5343:
5140:
4903:
4858:
4763:
4521:
4288:
4050:
3946:
3881:
3813:
3735:
3497:
3461:
3409:
3358:
3305:
3045:
2727:
2678:
2652:
2621:
2580:
2475:
2221:
1958:
1664:
1618:
1569:
1520:
1364:
1170:
1116:
1026:
930:
889:
844:
791:
620:
395:
252:
225:
205:
182:
117:
3552:
8593:Computational Fluid mechanics and Heat Transfer
8452:The 4th Conference of Iranian AeroSpace Society
7035:{\displaystyle p=\rho \left(\gamma -1\right)e,}
3462:{\displaystyle F\left(u\left(x,t\right)\right)}
8294:. The alternative form of van Albada limiter,
6894:(total energy). The total energy is given by,
6802:The equations above represent conservation of
183:{\displaystyle u_{t}+F_{x}\left(u\right)=0.\,}
8631:
8500:Finite Volume Methods for Hyperbolic Problems
8390:(advection upstream splitting method) scheme.
8:
8001:, is calculated from the equation of state.
7981:, is calculated from momentum, and pressure
6988:is required. One that suits our purpose is
6509:, was used to avoid spurious oscillations.
2634:section. The limiter is equal to zero when
8659:
8638:
8624:
8616:
8565:Principles of Computational Fluid Dynamics
8347:
8326:
8305:
8299:
8203:
8192:
8185:
8180:
8177:
8141:
8130:
8123:
8118:
8108:
8097:
8090:
8085:
8070:
8057:
8040:
8032:
8027:
8021:
8018:
8016:
7986:
7966:
7946:
7923:
7891:
7872:
7853:
7830:
7817:
7806:
7799:
7774:
7761:
7736:
7707:
7694:
7683:
7676:
7670:
7638:
7619:
7594:
7565:
7552:
7541:
7534:
7509:
7496:
7477:
7454:
7441:
7430:
7423:
7417:
7382:
7371:
7364:
7351:
7340:
7333:
7318:
7307:
7300:
7287:
7276:
7269:
7250:
7239:
7232:
7219:
7208:
7201:
7186:
7175:
7168:
7155:
7144:
7137:
7131:
7099:
7090:
7084:
7073:
7050:
6996:
6963:
6936:
6919:
6902:
6879:
6859:
6839:
6819:
6781:
6779:
6755:
6753:
6694:
6663:
6655:
6618:
6610:
6608:
6562:
6556:
6537:
6531:
6529:
6487:
6466:
6445:
6439:
6401:
6363:
6344:
6320:
6313:
6285:
6266:
6242:
6235:
6226:
6188:
6175:
6151:
6144:
6122:
6103:
6079:
6072:
6063:
6037:
6006:
5993:
5987:
5945:
5938:
5897:
5890:
5843:
5829:
5820:
5807:
5796:
5789:
5783:
5745:
5738:
5697:
5690:
5637:
5623:
5608:
5595:
5584:
5577:
5571:
5533:
5526:
5485:
5478:
5425:
5411:
5396:
5383:
5372:
5365:
5359:
5321:
5314:
5273:
5266:
5219:
5205:
5196:
5183:
5172:
5165:
5159:
5124:
5113:
5106:
5093:
5082:
5075:
5054:
5043:
5036:
5017:
5006:
4999:
4986:
4975:
4968:
4947:
4936:
4929:
4923:
4895:
4884:
4877:
4871:
4850:
4839:
4832:
4826:
4733:
4712:
4699:
4692:
4683:
4648:
4627:
4614:
4607:
4592:
4565:
4550:
4543:
4537:
4497:
4482:
4463:
4456:
4441:
4412:
4397:
4378:
4371:
4362:
4335:
4320:
4313:
4307:
4266:
4259:
4240:
4233:
4215:
4202:
4188:
4177:
4170:
4157:
4146:
4139:
4121:
4108:
4091:
4083:
4074:
4071:
4069:
4034:
4013:
3989:
3976:
3970:
3929:
3916:
3910:
3832:
3830:
3760:
3751:
3688:
3679:
3669:
3651:
3604:
3595:
3585:
3567:
3526:
3519:
3513:
3484:
3476:
3474:
3422:
3410:{\displaystyle a_{i\pm {\frac {1}{2}}}\ }
3392:
3385:
3379:
3344:
3331:
3325:
3284:
3273:
3266:
3253:
3242:
3235:
3214:
3207:
3185:
3174:
3167:
3143:
3132:
3125:
3094:
3085:
3074:
3067:
3061:
3024:
3013:
3006:
2993:
2982:
2975:
2954:
2947:
2925:
2914:
2907:
2883:
2872:
2865:
2834:
2825:
2814:
2807:
2801:
2719:
2710:
2700:
2694:
2665:
2639:
2609:
2596:
2566:
2547:
2529:
2516:
2509:
2500:
2494:
2459:
2440:
2421:
2398:
2385:
2376:
2366:
2342:
2329:
2304:
2275:
2262:
2253:
2243:
2237:
2199:
2180:
2155:
2126:
2113:
2104:
2094:
2076:
2057:
2038:
2015:
2002:
1993:
1983:
1977:
1942:
1933:
1923:
1910:
1901:
1891:
1876:
1867:
1857:
1844:
1835:
1825:
1807:
1798:
1788:
1775:
1766:
1756:
1741:
1732:
1722:
1709:
1700:
1690:
1684:
1656:
1647:
1637:
1631:
1610:
1601:
1591:
1585:
1561:
1552:
1542:
1536:
1501:
1492:
1482:
1469:
1460:
1450:
1432:
1419:
1405:
1397:
1388:
1385:
1383:
1341:
1332:
1322:
1298:
1289:
1279:
1254:
1241:
1227:
1219:
1210:
1207:
1205:
1156:
1143:
1138:
1136:
1100:
1081:
1056:
1046:
1040:
1004:
991:
966:
956:
950:
918:
908:
902:
877:
867:
861:
830:
817:
811:
761:
751:
723:
713:
688:
675:
661:
653:
644:
641:
639:
600:
587:
559:
540:
518:
499:
479:
462:
453:
430:
366:
342:
317:
304:
290:
282:
273:
270:
268:
245:
218:
198:
179:
156:
143:
137:
103:
90:
84:
7068:is equal to the ratio of specific heats
4904:{\displaystyle u_{i-{\frac {1}{2}}}^{*}}
4859:{\displaystyle u_{i+{\frac {1}{2}}}^{*}}
2622:{\displaystyle \phi \left(r_{i}\right)}
2755:, (Kurganov and Tadmor, 2000), is the
6981:represents specific internal energy.
6423:{\displaystyle \phi \left(r\right)\ }
7:
8889:Moving particle semi-implicit method
8800:Weighted essentially non-oscillatory
8591:Tannehill, John C., et al. (1997),
8738:Finite-difference frequency-domain
8063:
8041:
8022:
6569:
6559:
6544:
6534:
4805:Piecewise parabolic reconstruction
4726:
4641:
4490:
4405:
4208:
4114:
4092:
4075:
3867:
3835:
3795:
3763:
3712:
3654:
3628:
3570:
2743:Kurganov and Tadmor central scheme
1425:
1406:
1389:
1247:
1228:
1211:
681:
662:
645:
571:
310:
291:
274:
14:
9117:Numerical differential equations
8181:
8119:
8086:
8028:
6984:In order to close the system an
6656:
6611:
6563:
6538:
2793:The calculation is shown below:
1570:{\displaystyle F_{i\pm 1/2}^{*}}
213:represents a state variable and
9091:Method of fundamental solutions
8877:Smoothed-particle hydrodynamics
7794:
7529:
7264:
6733:
6654:
6305:
6136:
5031:
4774:Full details of the algorithm (
3947:{\displaystyle u_{t}+u_{x}=0\ }
1171:{\displaystyle \,u_{t}+u_{x}=0}
570:
8732:Alternating direction-implicit
8595:, 2nd Ed., Taylor and Francis.
8320:
8314:
6719:
6707:
6460:
6454:
1969:where, using downwind slopes:
612:
580:
21:partial differential equations
1:
8744:Finite-difference time-domain
8588:, Cambridge University Press.
8502:, Cambridge University Press.
6021:{\displaystyle u_{t}+u_{x}=0}
3359:{\displaystyle u_{t}+u_{x}=0}
2735:has been chosen, such as the
2728:{\displaystyle F_{i+1/2}^{*}}
1665:{\displaystyle u_{i-1/2}^{*}}
1619:{\displaystyle u_{i+1/2}^{*}}
845:{\displaystyle u_{t}+u_{x}=0}
118:{\displaystyle u_{t}+u_{x}=0}
9122:Computational fluid dynamics
8783:Advection upstream-splitting
3498:{\displaystyle {i},{i\pm 1}}
8794:Essentially non-oscillatory
8777:Monotonic upstream-centered
8536:J. Comput. Math. Phys. USSR
8434:Total variation diminishing
6513:Example: 1D Euler equations
2660:and is equal to unity when
1129:total variation diminishing
407:cells and used a 4th order
46:total variation diminishing
9138:
9054:Infinite difference method
8672:Forward-time central-space
8586:Computational Gas Dynamics
8584:Laney, Culbert B. (1998),
8563:Wesseling, Pieter (2001),
8272:lambda = 0.001069 (Δt/Δx).
6791:{\displaystyle {\mbox{F}}}
6773:is a vector of states and
6765:{\displaystyle {\mbox{U}}}
2737:Kurganov and Tadmor scheme
8957:Poincaré–Steklov operator
8716:Method of characteristics
7061:{\displaystyle \gamma \ }
6396:and the limiter function
6048:{\displaystyle \kappa \ }
931:{\displaystyle u_{i-1/2}}
890:{\displaystyle u_{i+1/2}}
8974:Tearing and interconnect
8968:Balancing by constraints
8612:(author: Arno Mayrhofer)
8459:Kurganov, Alexander and
8248:pressure left = 100000 ;
7918:Similarly, for momentum
6430:, is the same as above.
4795:parabolic reconstruction
9081:Computer-assisted proof
9059:Infinite element method
8847:Gradient discretisation
8498:Leveque, R. J. (2002).
8257:density right = 0.125 ;
8251:pressure right= 10000 ;
6799:is a vector of fluxes.
3372:local propagation speed
2653:{\displaystyle r\leq 0}
411:time integrator (RK4).
9069:Petrov–Galerkin method
8830:Discontinuous Galerkin
8492:Comm. Pure Appl. Math.
8419:High resolution scheme
8363:
8282:
8228:
8213:
8159:
7995:
7975:
7955:
7935:
7934:{\displaystyle \rho u}
7909:
7656:
7400:
7114:
7062:
7036:
6975:
6949:
6888:
6868:
6848:
6828:
6792:
6766:
6738:
6591:
6503:
6424:
6387:
6212:
6049:
6029:
6022:
5982:1D advective equation
5969:
5769:
5557:
5345:
5142:
4905:
4860:
4814:
4765:
4523:
4290:
4052:
3948:
3883:
3815:
3737:
3499:
3463:
3411:
3367:
3360:
3320:1D advective equation
3307:
3047:
2765:high-resolution scheme
2729:
2680:
2654:
2623:
2582:
2477:
2223:
1960:
1666:
1620:
1571:
1522:
1366:
1190:
1172:
1118:
1028:
932:
891:
853:
846:
806:1D advective equation
793:
622:
397:
254:
227:
207:
184:
126:
119:
79:1D advective equation
9049:Isogeometric analysis
8895:Material point method
8364:
8279:
8225:
8214:
8160:
7996:
7976:
7956:
7936:
7910:
7657:
7401:
7115:
7113:{\displaystyle \left}
7063:
7037:
6976:
6950:
6889:
6869:
6849:
6829:
6827:{\displaystyle \rho }
6793:
6767:
6739:
6592:
6504:
6425:
6388:
6213:
6050:
6023:
5981:
5970:
5770:
5558:
5346:
5143:
4906:
4861:
4812:
4766:
4524:
4291:
4053:
3949:
3884:
3816:
3738:
3500:
3464:
3412:
3361:
3319:
3308:
3048:
2730:
2681:
2655:
2624:
2583:
2478:
2224:
1961:
1667:
1621:
1572:
1531:The numerical fluxes
1523:
1367:
1188:
1173:
1119:
1029:
933:
892:
847:
805:
794:
623:
398:
255:
228:
208:
185:
120:
78:
71:Linear reconstruction
9086:Integrable algorithm
8912:Domain decomposition
8556:Toro, E. F. (1999),
8478:SIAM J. Sci. Comput.
8404:Finite volume method
8298:
8266:velocity right = 0 ;
8254:density left = 1.0 ;
8176:
8015:
7985:
7965:
7945:
7922:
7669:
7416:
7130:
7072:
7049:
6995:
6962:
6901:
6878:
6858:
6838:
6818:
6778:
6752:
6607:
6528:
6438:
6400:
6225:
6062:
6036:
5986:
5782:
5570:
5358:
5158:
4922:
4870:
4825:
4791:linear extrapolation
4536:
4306:
4068:
3969:
3909:
3829:
3750:
3512:
3473:
3421:
3378:
3324:
3060:
2800:
2693:
2664:
2638:
2595:
2493:
2236:
1976:
1683:
1630:
1584:
1535:
1382:
1204:
1135:
1039:
949:
901:
860:
810:
638:
429:
267:
244:
217:
197:
136:
83:
29:finite volume method
8930:Schwarz alternating
8853:Loubignac iteration
8577:Hirsch, C. (1990),
8263:velocity left = 0 ;
8208:
8146:
8113:
7941:, and total energy
7822:
7699:
7557:
7446:
7387:
7356:
7323:
7292:
7255:
7224:
7191:
7160:
6974:{\displaystyle e\ }
5812:
5600:
5388:
5188:
5129:
5098:
5059:
5022:
4991:
4952:
4900:
4855:
4193:
4162:
3900:Lax-Friedrichs flux
3693:
3609:
3289:
3258:
3190:
3148:
3090:
3029:
2998:
2930:
2888:
2830:
2777:central differences
2757:Nessyahu and Tadmor
2749:Kurganov and Tadmor
2747:A precursor to the
2724:
2679:{\displaystyle r=1}
2390:
2267:
2118:
2007:
1947:
1915:
1881:
1849:
1812:
1780:
1746:
1714:
1661:
1615:
1566:
1506:
1474:
1346:
1303:
65:Riemann-solver-free
9076:Validated numerics
8567:, Springer-Verlag.
8560:, Springer-Verlag.
8547:J. Fluid Mechanics
8516:Nessyahu, H. and
8359:
8292:van Albada limiter
8283:
8244:units) were used:
8229:
8209:
8179:
8155:
8117:
8084:
7991:
7971:
7951:
7931:
7905:
7795:
7672:
7652:
7530:
7419:
7396:
7360:
7329:
7296:
7265:
7228:
7197:
7164:
7133:
7110:
7058:
7032:
6971:
6945:
6884:
6864:
6854:(fluid velocity),
6844:
6824:
6788:
6786:
6762:
6760:
6734:
6724:
6648:
6587:
6499:
6420:
6383:
6208:
6045:
6030:
6018:
5965:
5785:
5765:
5573:
5553:
5361:
5341:
5161:
5138:
5102:
5071:
5032:
4995:
4964:
4925:
4901:
4873:
4856:
4828:
4815:
4761:
4519:
4286:
4166:
4135:
4048:
3944:
3879:
3811:
3733:
3665:
3581:
3495:
3459:
3407:
3368:
3356:
3303:
3262:
3231:
3163:
3121:
3063:
3043:
3002:
2971:
2903:
2861:
2803:
2725:
2696:
2676:
2650:
2619:
2578:
2473:
2362:
2239:
2219:
2090:
1979:
1956:
1919:
1887:
1853:
1821:
1784:
1752:
1718:
1686:
1662:
1633:
1616:
1587:
1567:
1538:
1518:
1478:
1446:
1362:
1318:
1275:
1191:
1168:
1114:
1024:
928:
887:
854:
842:
789:
618:
416:central difference
393:
250:
223:
203:
180:
127:
115:
9099:
9098:
9039:Immersed boundary
9032:Method of moments
8947:Neumann–Dirichlet
8940:abstract additive
8925:Fictitious domain
8869:Meshless/Meshfree
8753:
8752:
8655:Finite difference
8495:, VII, pp159–193.
8429:Sergei K. Godunov
8414:Godunov's theorem
8358:
8354:
8200:
8138:
8105:
8077:
8049:
7994:{\displaystyle p}
7974:{\displaystyle u}
7954:{\displaystyle E}
7814:
7691:
7549:
7438:
7379:
7348:
7315:
7284:
7247:
7216:
7183:
7152:
7057:
6986:equation of state
6970:
6927:
6887:{\displaystyle E}
6867:{\displaystyle p}
6847:{\displaystyle u}
6785:
6759:
6576:
6551:
6498:
6494:
6419:
6328:
6250:
6159:
6087:
6044:
5953:
5905:
5857:
5804:
5753:
5705:
5657:
5592:
5541:
5493:
5445:
5380:
5329:
5281:
5233:
5180:
5121:
5090:
5051:
5014:
4983:
4944:
4892:
4847:
4746:
4661:
4573:
4558:
4504:
4419:
4343:
4328:
4274:
4248:
4222:
4185:
4154:
4128:
4100:
3943:
3874:
3810:
3802:
3719:
3635:
3534:
3406:
3400:
3281:
3250:
3222:
3182:
3140:
3102:
3082:
3021:
2990:
2962:
2922:
2880:
2842:
2822:
2759:(NT) a staggered
2573:
1439:
1414:
1261:
1236:
1180:Godunov's theorem
695:
670:
529:
324:
299:
253:{\displaystyle i}
226:{\displaystyle F}
206:{\displaystyle u}
9129:
9044:Analytic element
9027:Boundary element
8920:Schur complement
8901:Particle-in-cell
8836:Spectral element
8660:
8640:
8633:
8626:
8617:
8522:J. Comput. Phys.
8465:J. Comput. Phys.
8368:
8366:
8365:
8360:
8356:
8355:
8353:
8352:
8351:
8335:
8327:
8313:
8312:
8269:duration =0.01 ;
8218:
8216:
8215:
8210:
8207:
8202:
8201:
8193:
8184:
8164:
8162:
8161:
8156:
8151:
8147:
8145:
8140:
8139:
8131:
8122:
8112:
8107:
8106:
8098:
8089:
8078:
8076:
8075:
8074:
8058:
8050:
8048:
8044:
8038:
8037:
8036:
8031:
8025:
8019:
8000:
7998:
7997:
7992:
7980:
7978:
7977:
7972:
7960:
7958:
7957:
7952:
7940:
7938:
7937:
7932:
7914:
7912:
7911:
7906:
7901:
7897:
7896:
7895:
7883:
7882:
7862:
7858:
7857:
7835:
7834:
7821:
7816:
7815:
7807:
7790:
7786:
7785:
7784:
7766:
7765:
7751:
7747:
7746:
7718:
7717:
7698:
7693:
7692:
7684:
7661:
7659:
7658:
7653:
7648:
7644:
7643:
7642:
7630:
7629:
7609:
7605:
7604:
7576:
7575:
7556:
7551:
7550:
7542:
7525:
7521:
7520:
7519:
7501:
7500:
7486:
7482:
7481:
7459:
7458:
7445:
7440:
7439:
7431:
7405:
7403:
7402:
7397:
7392:
7388:
7386:
7381:
7380:
7372:
7355:
7350:
7349:
7341:
7322:
7317:
7316:
7308:
7291:
7286:
7285:
7277:
7260:
7256:
7254:
7249:
7248:
7240:
7223:
7218:
7217:
7209:
7190:
7185:
7184:
7176:
7159:
7154:
7153:
7145:
7119:
7117:
7116:
7111:
7109:
7105:
7104:
7103:
7094:
7089:
7088:
7067:
7065:
7064:
7059:
7055:
7041:
7039:
7038:
7033:
7025:
7021:
6980:
6978:
6977:
6972:
6968:
6954:
6952:
6951:
6946:
6941:
6940:
6928:
6920:
6893:
6891:
6890:
6885:
6873:
6871:
6870:
6865:
6853:
6851:
6850:
6845:
6833:
6831:
6830:
6825:
6797:
6795:
6794:
6789:
6787:
6783:
6771:
6769:
6768:
6763:
6761:
6757:
6743:
6741:
6740:
6735:
6729:
6728:
6699:
6698:
6659:
6653:
6652:
6614:
6596:
6594:
6593:
6588:
6577:
6575:
6567:
6566:
6557:
6552:
6550:
6542:
6541:
6532:
6508:
6506:
6505:
6500:
6496:
6495:
6493:
6492:
6491:
6475:
6467:
6453:
6452:
6429:
6427:
6426:
6421:
6417:
6416:
6392:
6390:
6389:
6384:
6379:
6375:
6374:
6373:
6355:
6354:
6331:
6330:
6329:
6321:
6301:
6297:
6296:
6295:
6277:
6276:
6253:
6252:
6251:
6243:
6217:
6215:
6214:
6209:
6204:
6200:
6199:
6198:
6180:
6179:
6162:
6161:
6160:
6152:
6132:
6128:
6127:
6126:
6114:
6113:
6090:
6089:
6088:
6080:
6054:
6052:
6051:
6046:
6042:
6027:
6025:
6024:
6019:
6011:
6010:
5998:
5997:
5974:
5972:
5971:
5966:
5961:
5957:
5956:
5955:
5954:
5946:
5930:
5926:
5908:
5907:
5906:
5898:
5882:
5878:
5858:
5853:
5852:
5848:
5847:
5830:
5825:
5824:
5811:
5806:
5805:
5797:
5774:
5772:
5771:
5766:
5761:
5757:
5756:
5755:
5754:
5746:
5730:
5726:
5708:
5707:
5706:
5698:
5682:
5678:
5658:
5653:
5652:
5648:
5647:
5624:
5619:
5618:
5599:
5594:
5593:
5585:
5562:
5560:
5559:
5554:
5549:
5545:
5544:
5543:
5542:
5534:
5518:
5514:
5496:
5495:
5494:
5486:
5470:
5466:
5446:
5441:
5440:
5436:
5435:
5412:
5407:
5406:
5387:
5382:
5381:
5373:
5350:
5348:
5347:
5342:
5337:
5333:
5332:
5331:
5330:
5322:
5306:
5302:
5284:
5283:
5282:
5274:
5258:
5254:
5234:
5229:
5228:
5224:
5223:
5206:
5201:
5200:
5187:
5182:
5181:
5173:
5147:
5145:
5144:
5139:
5134:
5130:
5128:
5123:
5122:
5114:
5097:
5092:
5091:
5083:
5058:
5053:
5052:
5044:
5027:
5023:
5021:
5016:
5015:
5007:
4990:
4985:
4984:
4976:
4951:
4946:
4945:
4937:
4910:
4908:
4907:
4902:
4899:
4894:
4893:
4885:
4865:
4863:
4862:
4857:
4854:
4849:
4848:
4840:
4801:sections below.
4770:
4768:
4767:
4762:
4760:
4756:
4752:
4748:
4747:
4745:
4744:
4743:
4724:
4723:
4722:
4704:
4703:
4693:
4688:
4687:
4667:
4663:
4662:
4660:
4659:
4658:
4639:
4638:
4637:
4619:
4618:
4608:
4603:
4602:
4574:
4566:
4561:
4560:
4559:
4551:
4528:
4526:
4525:
4520:
4515:
4511:
4510:
4506:
4505:
4503:
4502:
4501:
4488:
4487:
4486:
4474:
4473:
4457:
4452:
4451:
4425:
4421:
4420:
4418:
4417:
4416:
4403:
4402:
4401:
4389:
4388:
4372:
4367:
4366:
4344:
4336:
4331:
4330:
4329:
4321:
4295:
4293:
4292:
4287:
4282:
4278:
4277:
4276:
4275:
4267:
4251:
4250:
4249:
4241:
4223:
4221:
4220:
4219:
4203:
4198:
4194:
4192:
4187:
4186:
4178:
4161:
4156:
4155:
4147:
4129:
4127:
4126:
4125:
4109:
4101:
4099:
4095:
4089:
4088:
4087:
4078:
4072:
4057:
4055:
4054:
4049:
4044:
4040:
4039:
4038:
4018:
4017:
4005:
3994:
3993:
3981:
3980:
3956:Superbee limiter
3953:
3951:
3950:
3945:
3941:
3934:
3933:
3921:
3920:
3888:
3886:
3885:
3880:
3875:
3873:
3865:
3864:
3860:
3859:
3833:
3820:
3818:
3817:
3812:
3808:
3807:
3803:
3801:
3793:
3792:
3788:
3787:
3761:
3742:
3740:
3739:
3734:
3732:
3728:
3724:
3720:
3718:
3710:
3709:
3705:
3704:
3692:
3687:
3683:
3652:
3640:
3636:
3634:
3626:
3625:
3621:
3620:
3608:
3603:
3599:
3568:
3548:
3537:
3536:
3535:
3527:
3504:
3502:
3501:
3496:
3494:
3480:
3468:
3466:
3465:
3460:
3458:
3454:
3453:
3449:
3416:
3414:
3413:
3408:
3404:
3403:
3402:
3401:
3393:
3365:
3363:
3362:
3357:
3349:
3348:
3336:
3335:
3312:
3310:
3309:
3304:
3299:
3295:
3294:
3290:
3288:
3283:
3282:
3274:
3257:
3252:
3251:
3243:
3225:
3224:
3223:
3215:
3199:
3195:
3194:
3189:
3184:
3183:
3175:
3152:
3147:
3142:
3141:
3133:
3103:
3095:
3089:
3084:
3083:
3075:
3052:
3050:
3049:
3044:
3039:
3035:
3034:
3030:
3028:
3023:
3022:
3014:
2997:
2992:
2991:
2983:
2965:
2964:
2963:
2955:
2939:
2935:
2934:
2929:
2924:
2923:
2915:
2892:
2887:
2882:
2881:
2873:
2843:
2835:
2829:
2824:
2823:
2815:
2734:
2732:
2731:
2726:
2723:
2718:
2714:
2685:
2683:
2682:
2677:
2659:
2657:
2656:
2651:
2628:
2626:
2625:
2620:
2618:
2614:
2613:
2587:
2585:
2584:
2579:
2574:
2572:
2571:
2570:
2558:
2557:
2541:
2540:
2539:
2521:
2520:
2510:
2505:
2504:
2482:
2480:
2479:
2474:
2469:
2465:
2464:
2463:
2451:
2450:
2430:
2426:
2425:
2403:
2402:
2389:
2384:
2380:
2358:
2354:
2353:
2352:
2334:
2333:
2319:
2315:
2314:
2286:
2285:
2266:
2261:
2257:
2228:
2226:
2225:
2220:
2215:
2211:
2210:
2209:
2191:
2190:
2170:
2166:
2165:
2137:
2136:
2117:
2112:
2108:
2086:
2082:
2081:
2080:
2068:
2067:
2047:
2043:
2042:
2020:
2019:
2006:
2001:
1997:
1965:
1963:
1962:
1957:
1952:
1948:
1946:
1941:
1937:
1914:
1909:
1905:
1880:
1875:
1871:
1848:
1843:
1839:
1817:
1813:
1811:
1806:
1802:
1779:
1774:
1770:
1745:
1740:
1736:
1713:
1708:
1704:
1671:
1669:
1668:
1663:
1660:
1655:
1651:
1625:
1623:
1622:
1617:
1614:
1609:
1605:
1576:
1574:
1573:
1568:
1565:
1560:
1556:
1527:
1525:
1524:
1519:
1511:
1507:
1505:
1500:
1496:
1473:
1468:
1464:
1440:
1438:
1437:
1436:
1420:
1415:
1413:
1409:
1403:
1402:
1401:
1392:
1386:
1371:
1369:
1368:
1363:
1355:
1351:
1350:
1345:
1340:
1336:
1307:
1302:
1297:
1293:
1262:
1260:
1259:
1258:
1242:
1237:
1235:
1231:
1225:
1224:
1223:
1214:
1208:
1177:
1175:
1174:
1169:
1161:
1160:
1148:
1147:
1123:
1121:
1120:
1115:
1110:
1106:
1105:
1104:
1092:
1091:
1065:
1064:
1060:
1033:
1031:
1030:
1025:
1020:
1016:
1015:
1014:
996:
995:
975:
974:
970:
937:
935:
934:
929:
927:
926:
922:
896:
894:
893:
888:
886:
885:
881:
851:
849:
848:
843:
835:
834:
822:
821:
798:
796:
795:
790:
779:
775:
774:
770:
769:
765:
736:
732:
731:
727:
696:
694:
693:
692:
676:
671:
669:
665:
659:
658:
657:
648:
642:
627:
625:
624:
619:
611:
610:
592:
591:
569:
565:
564:
563:
551:
550:
530:
528:
524:
523:
522:
510:
509:
489:
485:
484:
483:
463:
458:
457:
445:
402:
400:
399:
394:
386:
382:
381:
377:
376:
351:
347:
346:
325:
323:
322:
321:
305:
300:
298:
294:
288:
287:
286:
277:
271:
259:
257:
256:
251:
232:
230:
229:
224:
212:
210:
209:
204:
189:
187:
186:
181:
172:
161:
160:
148:
147:
124:
122:
121:
116:
108:
107:
95:
94:
54:Godunov's scheme
19:In the study of
9137:
9136:
9132:
9131:
9130:
9128:
9127:
9126:
9102:
9101:
9100:
9095:
9064:Galerkin method
9007:Method of lines
8984:
8952:Neumann–Neumann
8906:
8863:
8805:
8772:High-resolution
8749:
8720:
8682:
8649:
8644:
8602:
8581:, vol 2, Wiley.
8574:
8572:Further reading
8447:
8424:Method of lines
8400:
8343:
8336:
8328:
8301:
8296:
8295:
8174:
8173:
8083:
8079:
8066:
8062:
8039:
8026:
8020:
8013:
8012:
7983:
7982:
7963:
7962:
7943:
7942:
7920:
7919:
7887:
7868:
7867:
7863:
7849:
7845:
7826:
7770:
7757:
7756:
7752:
7732:
7728:
7703:
7667:
7666:
7634:
7615:
7614:
7610:
7590:
7586:
7561:
7505:
7492:
7491:
7487:
7473:
7469:
7450:
7414:
7413:
7328:
7324:
7196:
7192:
7128:
7127:
7120:for the fluid.
7095:
7080:
7079:
7075:
7070:
7069:
7047:
7046:
7011:
7007:
6993:
6992:
6960:
6959:
6932:
6899:
6898:
6876:
6875:
6874:(pressure) and
6856:
6855:
6836:
6835:
6816:
6815:
6776:
6775:
6750:
6749:
6723:
6722:
6701:
6700:
6690:
6678:
6677:
6664:
6647:
6646:
6640:
6639:
6630:
6629:
6619:
6605:
6604:
6568:
6558:
6543:
6533:
6526:
6525:
6519:Euler equations
6515:
6483:
6476:
6468:
6441:
6436:
6435:
6406:
6398:
6397:
6359:
6340:
6339:
6335:
6309:
6281:
6262:
6261:
6257:
6231:
6223:
6222:
6184:
6171:
6170:
6166:
6140:
6118:
6099:
6098:
6094:
6068:
6060:
6059:
6034:
6033:
6002:
5989:
5984:
5983:
5934:
5916:
5912:
5886:
5868:
5864:
5863:
5859:
5839:
5835:
5831:
5816:
5780:
5779:
5734:
5716:
5712:
5686:
5668:
5664:
5663:
5659:
5633:
5629:
5625:
5604:
5568:
5567:
5522:
5504:
5500:
5474:
5456:
5452:
5451:
5447:
5421:
5417:
5413:
5392:
5356:
5355:
5310:
5292:
5288:
5262:
5244:
5240:
5239:
5235:
5215:
5211:
5207:
5192:
5156:
5155:
5070:
5066:
4963:
4959:
4920:
4919:
4868:
4867:
4823:
4822:
4807:
4729:
4725:
4708:
4695:
4694:
4679:
4678:
4674:
4644:
4640:
4623:
4610:
4609:
4588:
4587:
4583:
4579:
4575:
4539:
4534:
4533:
4493:
4489:
4478:
4459:
4458:
4437:
4436:
4432:
4408:
4404:
4393:
4374:
4373:
4358:
4357:
4353:
4349:
4345:
4309:
4304:
4303:
4255:
4229:
4228:
4224:
4211:
4207:
4134:
4130:
4117:
4113:
4090:
4079:
4073:
4066:
4065:
4030:
4023:
4019:
4009:
3995:
3985:
3972:
3967:
3966:
3925:
3912:
3907:
3906:
3866:
3849:
3845:
3841:
3834:
3827:
3826:
3823:spectral radius
3821:represents the
3794:
3777:
3773:
3769:
3762:
3756:
3748:
3747:
3711:
3694:
3664:
3660:
3653:
3647:
3627:
3610:
3580:
3576:
3569:
3563:
3559:
3555:
3538:
3515:
3510:
3509:
3471:
3470:
3439:
3435:
3431:
3427:
3419:
3418:
3381:
3376:
3375:
3340:
3327:
3322:
3321:
3230:
3226:
3203:
3159:
3117:
3113:
3109:
3108:
3104:
3058:
3057:
2970:
2966:
2943:
2899:
2857:
2853:
2849:
2848:
2844:
2798:
2797:
2745:
2691:
2690:
2662:
2661:
2636:
2635:
2605:
2601:
2593:
2592:
2562:
2543:
2542:
2525:
2512:
2511:
2496:
2491:
2490:
2455:
2436:
2435:
2431:
2417:
2413:
2394:
2338:
2325:
2324:
2320:
2300:
2296:
2271:
2234:
2233:
2195:
2176:
2175:
2171:
2151:
2147:
2122:
2072:
2053:
2052:
2048:
2034:
2030:
2011:
1974:
1973:
1886:
1882:
1751:
1747:
1681:
1680:
1628:
1627:
1582:
1581:
1533:
1532:
1445:
1441:
1428:
1424:
1404:
1393:
1387:
1380:
1379:
1314:
1271:
1267:
1263:
1250:
1246:
1226:
1215:
1209:
1202:
1201:
1152:
1139:
1133:
1132:
1096:
1077:
1076:
1072:
1042:
1037:
1036:
1000:
987:
986:
982:
952:
947:
946:
904:
899:
898:
863:
858:
857:
826:
813:
808:
807:
747:
743:
709:
705:
701:
697:
684:
680:
660:
649:
643:
636:
635:
596:
583:
555:
536:
535:
531:
514:
495:
494:
490:
475:
468:
464:
449:
435:
427:
426:
418:scheme that is
362:
358:
338:
334:
330:
326:
313:
309:
289:
278:
272:
265:
264:
242:
241:
215:
214:
195:
194:
162:
152:
139:
134:
133:
99:
86:
81:
80:
73:
17:
12:
11:
5:
9135:
9133:
9125:
9124:
9119:
9114:
9112:Fluid dynamics
9104:
9103:
9097:
9096:
9094:
9093:
9088:
9083:
9078:
9073:
9072:
9071:
9061:
9056:
9051:
9046:
9041:
9036:
9035:
9034:
9024:
9019:
9014:
9009:
9004:
9001:Pseudospectral
8998:
8992:
8990:
8986:
8985:
8983:
8982:
8977:
8971:
8965:
8959:
8954:
8949:
8944:
8943:
8942:
8937:
8927:
8922:
8916:
8914:
8908:
8907:
8905:
8904:
8898:
8892:
8886:
8880:
8873:
8871:
8865:
8864:
8862:
8861:
8855:
8850:
8844:
8839:
8833:
8827:
8821:
8815:
8813:
8811:Finite element
8807:
8806:
8804:
8803:
8797:
8791:
8789:Riemann solver
8786:
8780:
8774:
8769:
8763:
8761:
8755:
8754:
8751:
8750:
8748:
8747:
8741:
8735:
8728:
8726:
8722:
8721:
8719:
8718:
8713:
8708:
8703:
8698:
8696:Lax–Friedrichs
8692:
8690:
8684:
8683:
8681:
8680:
8678:Crank–Nicolson
8675:
8668:
8666:
8657:
8651:
8650:
8645:
8643:
8642:
8635:
8628:
8620:
8614:
8613:
8601:
8600:External links
8598:
8597:
8596:
8589:
8582:
8573:
8570:
8569:
8568:
8561:
8554:
8543:
8532:
8514:
8503:
8496:
8487:
8474:
8457:
8446:
8443:
8442:
8441:
8439:Sod shock tube
8436:
8431:
8426:
8421:
8416:
8411:
8406:
8399:
8396:
8392:
8391:
8381:
8350:
8346:
8342:
8339:
8334:
8331:
8325:
8322:
8319:
8316:
8311:
8308:
8304:
8274:
8273:
8270:
8267:
8264:
8261:
8258:
8255:
8252:
8249:
8206:
8199:
8196:
8191:
8188:
8183:
8166:
8165:
8154:
8150:
8144:
8137:
8134:
8129:
8126:
8121:
8116:
8111:
8104:
8101:
8096:
8093:
8088:
8082:
8073:
8069:
8065:
8061:
8056:
8053:
8047:
8043:
8035:
8030:
8024:
7990:
7970:
7950:
7930:
7927:
7916:
7915:
7904:
7900:
7894:
7890:
7886:
7881:
7878:
7875:
7871:
7866:
7861:
7856:
7852:
7848:
7844:
7841:
7838:
7833:
7829:
7825:
7820:
7813:
7810:
7805:
7802:
7798:
7793:
7789:
7783:
7780:
7777:
7773:
7769:
7764:
7760:
7755:
7750:
7745:
7742:
7739:
7735:
7731:
7727:
7724:
7721:
7716:
7713:
7710:
7706:
7702:
7697:
7690:
7687:
7682:
7679:
7675:
7663:
7662:
7651:
7647:
7641:
7637:
7633:
7628:
7625:
7622:
7618:
7613:
7608:
7603:
7600:
7597:
7593:
7589:
7585:
7582:
7579:
7574:
7571:
7568:
7564:
7560:
7555:
7548:
7545:
7540:
7537:
7533:
7528:
7524:
7518:
7515:
7512:
7508:
7504:
7499:
7495:
7490:
7485:
7480:
7476:
7472:
7468:
7465:
7462:
7457:
7453:
7449:
7444:
7437:
7434:
7429:
7426:
7422:
7407:
7406:
7395:
7391:
7385:
7378:
7375:
7370:
7367:
7363:
7359:
7354:
7347:
7344:
7339:
7336:
7332:
7327:
7321:
7314:
7311:
7306:
7303:
7299:
7295:
7290:
7283:
7280:
7275:
7272:
7268:
7263:
7259:
7253:
7246:
7243:
7238:
7235:
7231:
7227:
7222:
7215:
7212:
7207:
7204:
7200:
7195:
7189:
7182:
7179:
7174:
7171:
7167:
7163:
7158:
7151:
7148:
7143:
7140:
7136:
7108:
7102:
7098:
7093:
7087:
7083:
7078:
7054:
7043:
7042:
7031:
7028:
7024:
7020:
7017:
7014:
7010:
7006:
7003:
7000:
6967:
6956:
6955:
6944:
6939:
6935:
6931:
6926:
6923:
6918:
6915:
6912:
6909:
6906:
6883:
6863:
6843:
6823:
6745:
6744:
6732:
6727:
6721:
6718:
6715:
6712:
6709:
6706:
6703:
6702:
6697:
6693:
6689:
6686:
6683:
6680:
6679:
6676:
6673:
6670:
6669:
6667:
6662:
6658:
6651:
6645:
6642:
6641:
6638:
6635:
6632:
6631:
6628:
6625:
6624:
6622:
6617:
6613:
6598:
6597:
6586:
6583:
6580:
6574:
6571:
6565:
6561:
6555:
6549:
6546:
6540:
6536:
6514:
6511:
6490:
6486:
6482:
6479:
6474:
6471:
6465:
6462:
6459:
6456:
6451:
6448:
6444:
6415:
6412:
6409:
6405:
6394:
6393:
6382:
6378:
6372:
6369:
6366:
6362:
6358:
6353:
6350:
6347:
6343:
6338:
6334:
6327:
6324:
6319:
6316:
6312:
6308:
6304:
6300:
6294:
6291:
6288:
6284:
6280:
6275:
6272:
6269:
6265:
6260:
6256:
6249:
6246:
6241:
6238:
6234:
6230:
6219:
6218:
6207:
6203:
6197:
6194:
6191:
6187:
6183:
6178:
6174:
6169:
6165:
6158:
6155:
6150:
6147:
6143:
6139:
6135:
6131:
6125:
6121:
6117:
6112:
6109:
6106:
6102:
6097:
6093:
6086:
6083:
6078:
6075:
6071:
6067:
6041:
6017:
6014:
6009:
6005:
6001:
5996:
5992:
5976:
5975:
5964:
5960:
5952:
5949:
5944:
5941:
5937:
5933:
5929:
5925:
5922:
5919:
5915:
5911:
5904:
5901:
5896:
5893:
5889:
5885:
5881:
5877:
5874:
5871:
5867:
5862:
5856:
5851:
5846:
5842:
5838:
5834:
5828:
5823:
5819:
5815:
5810:
5803:
5800:
5795:
5792:
5788:
5776:
5775:
5764:
5760:
5752:
5749:
5744:
5741:
5737:
5733:
5729:
5725:
5722:
5719:
5715:
5711:
5704:
5701:
5696:
5693:
5689:
5685:
5681:
5677:
5674:
5671:
5667:
5662:
5656:
5651:
5646:
5643:
5640:
5636:
5632:
5628:
5622:
5617:
5614:
5611:
5607:
5603:
5598:
5591:
5588:
5583:
5580:
5576:
5564:
5563:
5552:
5548:
5540:
5537:
5532:
5529:
5525:
5521:
5517:
5513:
5510:
5507:
5503:
5499:
5492:
5489:
5484:
5481:
5477:
5473:
5469:
5465:
5462:
5459:
5455:
5450:
5444:
5439:
5434:
5431:
5428:
5424:
5420:
5416:
5410:
5405:
5402:
5399:
5395:
5391:
5386:
5379:
5376:
5371:
5368:
5364:
5352:
5351:
5340:
5336:
5328:
5325:
5320:
5317:
5313:
5309:
5305:
5301:
5298:
5295:
5291:
5287:
5280:
5277:
5272:
5269:
5265:
5261:
5257:
5253:
5250:
5247:
5243:
5238:
5232:
5227:
5222:
5218:
5214:
5210:
5204:
5199:
5195:
5191:
5186:
5179:
5176:
5171:
5168:
5164:
5149:
5148:
5137:
5133:
5127:
5120:
5117:
5112:
5109:
5105:
5101:
5096:
5089:
5086:
5081:
5078:
5074:
5069:
5065:
5062:
5057:
5050:
5047:
5042:
5039:
5035:
5030:
5026:
5020:
5013:
5010:
5005:
5002:
4998:
4994:
4989:
4982:
4979:
4974:
4971:
4967:
4962:
4958:
4955:
4950:
4943:
4940:
4935:
4932:
4928:
4898:
4891:
4888:
4883:
4880:
4876:
4853:
4846:
4843:
4838:
4835:
4831:
4806:
4803:
4799:Euler equation
4772:
4771:
4759:
4755:
4751:
4742:
4739:
4736:
4732:
4728:
4721:
4718:
4715:
4711:
4707:
4702:
4698:
4691:
4686:
4682:
4677:
4673:
4670:
4666:
4657:
4654:
4651:
4647:
4643:
4636:
4633:
4630:
4626:
4622:
4617:
4613:
4606:
4601:
4598:
4595:
4591:
4586:
4582:
4578:
4572:
4569:
4564:
4557:
4554:
4549:
4546:
4542:
4530:
4529:
4518:
4514:
4509:
4500:
4496:
4492:
4485:
4481:
4477:
4472:
4469:
4466:
4462:
4455:
4450:
4447:
4444:
4440:
4435:
4431:
4428:
4424:
4415:
4411:
4407:
4400:
4396:
4392:
4387:
4384:
4381:
4377:
4370:
4365:
4361:
4356:
4352:
4348:
4342:
4339:
4334:
4327:
4324:
4319:
4316:
4312:
4297:
4296:
4285:
4281:
4273:
4270:
4265:
4262:
4258:
4254:
4247:
4244:
4239:
4236:
4232:
4227:
4218:
4214:
4210:
4206:
4201:
4197:
4191:
4184:
4181:
4176:
4173:
4169:
4165:
4160:
4153:
4150:
4145:
4142:
4138:
4133:
4124:
4120:
4116:
4112:
4107:
4104:
4098:
4094:
4086:
4082:
4077:
4059:
4058:
4047:
4043:
4037:
4033:
4029:
4026:
4022:
4016:
4012:
4008:
4004:
4001:
3998:
3992:
3988:
3984:
3979:
3975:
3940:
3937:
3932:
3928:
3924:
3919:
3915:
3878:
3872:
3869:
3863:
3858:
3855:
3852:
3848:
3844:
3840:
3837:
3806:
3800:
3797:
3791:
3786:
3783:
3780:
3776:
3772:
3768:
3765:
3759:
3755:
3744:
3743:
3731:
3727:
3723:
3717:
3714:
3708:
3703:
3700:
3697:
3691:
3686:
3682:
3678:
3675:
3672:
3668:
3663:
3659:
3656:
3650:
3646:
3643:
3639:
3633:
3630:
3624:
3619:
3616:
3613:
3607:
3602:
3598:
3594:
3591:
3588:
3584:
3579:
3575:
3572:
3566:
3562:
3558:
3554:
3551:
3547:
3544:
3541:
3533:
3530:
3525:
3522:
3518:
3493:
3490:
3487:
3483:
3479:
3457:
3452:
3448:
3445:
3442:
3438:
3434:
3430:
3426:
3399:
3396:
3391:
3388:
3384:
3355:
3352:
3347:
3343:
3339:
3334:
3330:
3314:
3313:
3302:
3298:
3293:
3287:
3280:
3277:
3272:
3269:
3265:
3261:
3256:
3249:
3246:
3241:
3238:
3234:
3229:
3221:
3218:
3213:
3210:
3206:
3202:
3198:
3193:
3188:
3181:
3178:
3173:
3170:
3166:
3162:
3158:
3155:
3151:
3146:
3139:
3136:
3131:
3128:
3124:
3120:
3116:
3112:
3107:
3101:
3098:
3093:
3088:
3081:
3078:
3073:
3070:
3066:
3054:
3053:
3042:
3038:
3033:
3027:
3020:
3017:
3012:
3009:
3005:
3001:
2996:
2989:
2986:
2981:
2978:
2974:
2969:
2961:
2958:
2953:
2950:
2946:
2942:
2938:
2933:
2928:
2921:
2918:
2913:
2910:
2906:
2902:
2898:
2895:
2891:
2886:
2879:
2876:
2871:
2868:
2864:
2860:
2856:
2852:
2847:
2841:
2838:
2833:
2828:
2821:
2818:
2813:
2810:
2806:
2784:fully discrete
2761:central scheme
2753:central scheme
2744:
2741:
2722:
2717:
2713:
2709:
2706:
2703:
2699:
2675:
2672:
2669:
2649:
2646:
2643:
2617:
2612:
2608:
2604:
2600:
2589:
2588:
2577:
2569:
2565:
2561:
2556:
2553:
2550:
2546:
2538:
2535:
2532:
2528:
2524:
2519:
2515:
2508:
2503:
2499:
2484:
2483:
2472:
2468:
2462:
2458:
2454:
2449:
2446:
2443:
2439:
2434:
2429:
2424:
2420:
2416:
2412:
2409:
2406:
2401:
2397:
2393:
2388:
2383:
2379:
2375:
2372:
2369:
2365:
2361:
2357:
2351:
2348:
2345:
2341:
2337:
2332:
2328:
2323:
2318:
2313:
2310:
2307:
2303:
2299:
2295:
2292:
2289:
2284:
2281:
2278:
2274:
2270:
2265:
2260:
2256:
2252:
2249:
2246:
2242:
2230:
2229:
2218:
2214:
2208:
2205:
2202:
2198:
2194:
2189:
2186:
2183:
2179:
2174:
2169:
2164:
2161:
2158:
2154:
2150:
2146:
2143:
2140:
2135:
2132:
2129:
2125:
2121:
2116:
2111:
2107:
2103:
2100:
2097:
2093:
2089:
2085:
2079:
2075:
2071:
2066:
2063:
2060:
2056:
2051:
2046:
2041:
2037:
2033:
2029:
2026:
2023:
2018:
2014:
2010:
2005:
2000:
1996:
1992:
1989:
1986:
1982:
1967:
1966:
1955:
1951:
1945:
1940:
1936:
1932:
1929:
1926:
1922:
1918:
1913:
1908:
1904:
1900:
1897:
1894:
1890:
1885:
1879:
1874:
1870:
1866:
1863:
1860:
1856:
1852:
1847:
1842:
1838:
1834:
1831:
1828:
1824:
1820:
1816:
1810:
1805:
1801:
1797:
1794:
1791:
1787:
1783:
1778:
1773:
1769:
1765:
1762:
1759:
1755:
1750:
1744:
1739:
1735:
1731:
1728:
1725:
1721:
1717:
1712:
1707:
1703:
1699:
1696:
1693:
1689:
1659:
1654:
1650:
1646:
1643:
1640:
1636:
1613:
1608:
1604:
1600:
1597:
1594:
1590:
1564:
1559:
1555:
1551:
1548:
1545:
1541:
1529:
1528:
1517:
1514:
1510:
1504:
1499:
1495:
1491:
1488:
1485:
1481:
1477:
1472:
1467:
1463:
1459:
1456:
1453:
1449:
1444:
1435:
1431:
1427:
1423:
1418:
1412:
1408:
1400:
1396:
1391:
1373:
1372:
1361:
1358:
1354:
1349:
1344:
1339:
1335:
1331:
1328:
1325:
1321:
1317:
1313:
1310:
1306:
1301:
1296:
1292:
1288:
1285:
1282:
1278:
1274:
1270:
1266:
1257:
1253:
1249:
1245:
1240:
1234:
1230:
1222:
1218:
1213:
1167:
1164:
1159:
1155:
1151:
1146:
1142:
1125:
1124:
1113:
1109:
1103:
1099:
1095:
1090:
1087:
1084:
1080:
1075:
1071:
1068:
1063:
1059:
1055:
1052:
1049:
1045:
1034:
1023:
1019:
1013:
1010:
1007:
1003:
999:
994:
990:
985:
981:
978:
973:
969:
965:
962:
959:
955:
925:
921:
917:
914:
911:
907:
884:
880:
876:
873:
870:
866:
841:
838:
833:
829:
825:
820:
816:
800:
799:
788:
785:
782:
778:
773:
768:
764:
760:
757:
754:
750:
746:
742:
739:
735:
730:
726:
722:
719:
716:
712:
708:
704:
700:
691:
687:
683:
679:
674:
668:
664:
656:
652:
647:
629:
628:
617:
614:
609:
606:
603:
599:
595:
590:
586:
582:
579:
576:
573:
568:
562:
558:
554:
549:
546:
543:
539:
534:
527:
521:
517:
513:
508:
505:
502:
498:
493:
488:
482:
478:
474:
471:
467:
461:
456:
452:
448:
444:
441:
438:
434:
404:
403:
392:
389:
385:
380:
375:
372:
369:
365:
361:
357:
354:
350:
345:
341:
337:
333:
329:
320:
316:
312:
308:
303:
297:
293:
285:
281:
276:
249:
222:
202:
191:
190:
178:
175:
171:
168:
165:
159:
155:
151:
146:
142:
114:
111:
106:
102:
98:
93:
89:
72:
69:
60:Riemann solver
15:
13:
10:
9:
6:
4:
3:
2:
9134:
9123:
9120:
9118:
9115:
9113:
9110:
9109:
9107:
9092:
9089:
9087:
9084:
9082:
9079:
9077:
9074:
9070:
9067:
9066:
9065:
9062:
9060:
9057:
9055:
9052:
9050:
9047:
9045:
9042:
9040:
9037:
9033:
9030:
9029:
9028:
9025:
9023:
9020:
9018:
9015:
9013:
9010:
9008:
9005:
9002:
8999:
8997:
8994:
8993:
8991:
8987:
8981:
8978:
8975:
8972:
8969:
8966:
8963:
8960:
8958:
8955:
8953:
8950:
8948:
8945:
8941:
8938:
8936:
8933:
8932:
8931:
8928:
8926:
8923:
8921:
8918:
8917:
8915:
8913:
8909:
8902:
8899:
8896:
8893:
8890:
8887:
8884:
8881:
8878:
8875:
8874:
8872:
8870:
8866:
8859:
8856:
8854:
8851:
8848:
8845:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8820:
8817:
8816:
8814:
8812:
8808:
8801:
8798:
8795:
8792:
8790:
8787:
8784:
8781:
8778:
8775:
8773:
8770:
8768:
8765:
8764:
8762:
8760:
8759:Finite volume
8756:
8745:
8742:
8739:
8736:
8733:
8730:
8729:
8727:
8723:
8717:
8714:
8712:
8709:
8707:
8704:
8702:
8699:
8697:
8694:
8693:
8691:
8689:
8685:
8679:
8676:
8673:
8670:
8669:
8667:
8665:
8661:
8658:
8656:
8652:
8648:
8641:
8636:
8634:
8629:
8627:
8622:
8621:
8618:
8611:
8607:
8604:
8603:
8599:
8594:
8590:
8587:
8583:
8580:
8576:
8575:
8571:
8566:
8562:
8559:
8555:
8552:
8548:
8544:
8541:
8537:
8533:
8530:
8527:
8523:
8519:
8515:
8512:
8508:
8507:J. Com. Phys.
8504:
8501:
8497:
8494:
8493:
8488:
8486:
8484:, 1461–1488.
8483:
8479:
8475:
8473:
8470:
8466:
8462:
8458:
8456:
8453:
8449:
8448:
8444:
8440:
8437:
8435:
8432:
8430:
8427:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8401:
8397:
8395:
8389:
8386:
8382:
8379:
8375:
8374:
8373:
8370:
8348:
8344:
8340:
8337:
8332:
8329:
8323:
8317:
8309:
8306:
8302:
8293:
8288:
8278:
8271:
8268:
8265:
8262:
8260:length = 20 ;
8259:
8256:
8253:
8250:
8247:
8246:
8245:
8243:
8239:
8238:Ospre limiter
8234:
8224:
8220:
8204:
8197:
8194:
8189:
8186:
8169:
8152:
8148:
8142:
8135:
8132:
8127:
8124:
8114:
8109:
8102:
8099:
8094:
8091:
8080:
8071:
8067:
8059:
8054:
8051:
8045:
8033:
8011:
8010:
8009:
8007:
8002:
7988:
7968:
7948:
7928:
7925:
7902:
7898:
7892:
7888:
7884:
7879:
7876:
7873:
7869:
7864:
7859:
7854:
7850:
7846:
7842:
7839:
7836:
7831:
7827:
7823:
7818:
7811:
7808:
7803:
7800:
7796:
7791:
7787:
7781:
7778:
7775:
7771:
7767:
7762:
7758:
7753:
7748:
7743:
7740:
7737:
7733:
7729:
7725:
7722:
7719:
7714:
7711:
7708:
7704:
7700:
7695:
7688:
7685:
7680:
7677:
7673:
7665:
7664:
7649:
7645:
7639:
7635:
7631:
7626:
7623:
7620:
7616:
7611:
7606:
7601:
7598:
7595:
7591:
7587:
7583:
7580:
7577:
7572:
7569:
7566:
7562:
7558:
7553:
7546:
7543:
7538:
7535:
7531:
7526:
7522:
7516:
7513:
7510:
7506:
7502:
7497:
7493:
7488:
7483:
7478:
7474:
7470:
7466:
7463:
7460:
7455:
7451:
7447:
7442:
7435:
7432:
7427:
7424:
7420:
7412:
7411:
7410:
7393:
7389:
7383:
7376:
7373:
7368:
7365:
7361:
7357:
7352:
7345:
7342:
7337:
7334:
7330:
7325:
7319:
7312:
7309:
7304:
7301:
7297:
7293:
7288:
7281:
7278:
7273:
7270:
7266:
7261:
7257:
7251:
7244:
7241:
7236:
7233:
7229:
7225:
7220:
7213:
7210:
7205:
7202:
7198:
7193:
7187:
7180:
7177:
7172:
7169:
7165:
7161:
7156:
7149:
7146:
7141:
7138:
7134:
7126:
7125:
7124:
7121:
7106:
7100:
7096:
7091:
7085:
7081:
7076:
7052:
7029:
7026:
7022:
7018:
7015:
7012:
7008:
7004:
7001:
6998:
6991:
6990:
6989:
6987:
6982:
6965:
6942:
6937:
6933:
6929:
6924:
6921:
6916:
6913:
6910:
6907:
6904:
6897:
6896:
6895:
6881:
6861:
6841:
6821:
6813:
6809:
6805:
6800:
6798:
6772:
6730:
6725:
6716:
6713:
6710:
6704:
6695:
6691:
6687:
6684:
6681:
6674:
6671:
6665:
6660:
6649:
6643:
6636:
6633:
6626:
6620:
6615:
6603:
6602:
6601:
6584:
6581:
6578:
6572:
6553:
6547:
6524:
6523:
6522:
6520:
6512:
6510:
6488:
6484:
6480:
6477:
6472:
6469:
6463:
6457:
6449:
6446:
6442:
6431:
6413:
6410:
6407:
6403:
6380:
6376:
6370:
6367:
6364:
6360:
6356:
6351:
6348:
6345:
6341:
6336:
6332:
6325:
6322:
6317:
6314:
6310:
6306:
6302:
6298:
6292:
6289:
6286:
6282:
6278:
6273:
6270:
6267:
6263:
6258:
6254:
6247:
6244:
6239:
6236:
6232:
6228:
6221:
6220:
6205:
6201:
6195:
6192:
6189:
6185:
6181:
6176:
6172:
6167:
6163:
6156:
6153:
6148:
6145:
6141:
6137:
6133:
6129:
6123:
6119:
6115:
6110:
6107:
6104:
6100:
6095:
6091:
6084:
6081:
6076:
6073:
6069:
6065:
6058:
6057:
6056:
6039:
6015:
6012:
6007:
6003:
5999:
5994:
5990:
5980:
5962:
5958:
5950:
5947:
5942:
5939:
5935:
5931:
5927:
5923:
5920:
5917:
5913:
5909:
5902:
5899:
5894:
5891:
5887:
5883:
5879:
5875:
5872:
5869:
5865:
5860:
5854:
5849:
5844:
5840:
5836:
5832:
5826:
5821:
5817:
5813:
5808:
5801:
5798:
5793:
5790:
5786:
5778:
5777:
5762:
5758:
5750:
5747:
5742:
5739:
5735:
5731:
5727:
5723:
5720:
5717:
5713:
5709:
5702:
5699:
5694:
5691:
5687:
5683:
5679:
5675:
5672:
5669:
5665:
5660:
5654:
5649:
5644:
5641:
5638:
5634:
5630:
5626:
5620:
5615:
5612:
5609:
5605:
5601:
5596:
5589:
5586:
5581:
5578:
5574:
5566:
5565:
5550:
5546:
5538:
5535:
5530:
5527:
5523:
5519:
5515:
5511:
5508:
5505:
5501:
5497:
5490:
5487:
5482:
5479:
5475:
5471:
5467:
5463:
5460:
5457:
5453:
5448:
5442:
5437:
5432:
5429:
5426:
5422:
5418:
5414:
5408:
5403:
5400:
5397:
5393:
5389:
5384:
5377:
5374:
5369:
5366:
5362:
5354:
5353:
5338:
5334:
5326:
5323:
5318:
5315:
5311:
5307:
5303:
5299:
5296:
5293:
5289:
5285:
5278:
5275:
5270:
5267:
5263:
5259:
5255:
5251:
5248:
5245:
5241:
5236:
5230:
5225:
5220:
5216:
5212:
5208:
5202:
5197:
5193:
5189:
5184:
5177:
5174:
5169:
5166:
5162:
5154:
5153:
5152:
5135:
5131:
5125:
5118:
5115:
5110:
5107:
5103:
5099:
5094:
5087:
5084:
5079:
5076:
5072:
5067:
5063:
5060:
5055:
5048:
5045:
5040:
5037:
5033:
5028:
5024:
5018:
5011:
5008:
5003:
5000:
4996:
4992:
4987:
4980:
4977:
4972:
4969:
4965:
4960:
4956:
4953:
4948:
4941:
4938:
4933:
4930:
4926:
4918:
4917:
4916:
4914:
4896:
4889:
4886:
4881:
4878:
4874:
4851:
4844:
4841:
4836:
4833:
4829:
4819:
4811:
4804:
4802:
4800:
4796:
4792:
4788:
4784:
4781:
4780:semi-discrete
4777:
4757:
4753:
4749:
4740:
4737:
4734:
4730:
4719:
4716:
4713:
4709:
4705:
4700:
4696:
4689:
4684:
4680:
4675:
4671:
4668:
4664:
4655:
4652:
4649:
4645:
4634:
4631:
4628:
4624:
4620:
4615:
4611:
4604:
4599:
4596:
4593:
4589:
4584:
4580:
4576:
4570:
4567:
4562:
4555:
4552:
4547:
4544:
4540:
4532:
4531:
4516:
4512:
4507:
4498:
4494:
4483:
4479:
4475:
4470:
4467:
4464:
4460:
4453:
4448:
4445:
4442:
4438:
4433:
4429:
4426:
4422:
4413:
4409:
4398:
4394:
4390:
4385:
4382:
4379:
4375:
4368:
4363:
4359:
4354:
4350:
4346:
4340:
4337:
4332:
4325:
4322:
4317:
4314:
4310:
4302:
4301:
4300:
4283:
4279:
4271:
4268:
4263:
4260:
4256:
4252:
4245:
4242:
4237:
4234:
4230:
4225:
4216:
4212:
4204:
4199:
4195:
4189:
4182:
4179:
4174:
4171:
4167:
4163:
4158:
4151:
4148:
4143:
4140:
4136:
4131:
4122:
4118:
4110:
4105:
4102:
4096:
4084:
4080:
4064:
4063:
4062:
4045:
4041:
4035:
4031:
4027:
4024:
4020:
4014:
4010:
4006:
4002:
3999:
3996:
3990:
3986:
3982:
3977:
3973:
3965:
3964:
3963:
3960:
3957:
3938:
3935:
3930:
3926:
3922:
3917:
3913:
3903:
3901:
3896:
3894:
3891:Beyond these
3889:
3876:
3870:
3861:
3856:
3853:
3850:
3846:
3842:
3838:
3824:
3804:
3798:
3789:
3784:
3781:
3778:
3774:
3770:
3766:
3757:
3753:
3729:
3725:
3721:
3715:
3706:
3701:
3698:
3695:
3689:
3684:
3680:
3676:
3673:
3670:
3666:
3661:
3657:
3648:
3644:
3641:
3637:
3631:
3622:
3617:
3614:
3611:
3605:
3600:
3596:
3592:
3589:
3586:
3582:
3577:
3573:
3564:
3560:
3556:
3549:
3545:
3542:
3539:
3531:
3528:
3523:
3520:
3516:
3508:
3507:
3506:
3491:
3488:
3485:
3481:
3477:
3455:
3450:
3446:
3443:
3440:
3436:
3432:
3428:
3424:
3397:
3394:
3389:
3386:
3382:
3373:
3353:
3350:
3345:
3341:
3337:
3332:
3328:
3318:
3300:
3296:
3291:
3285:
3278:
3275:
3270:
3267:
3263:
3259:
3254:
3247:
3244:
3239:
3236:
3232:
3227:
3219:
3216:
3211:
3208:
3204:
3200:
3196:
3191:
3186:
3179:
3176:
3171:
3168:
3164:
3160:
3156:
3153:
3149:
3144:
3137:
3134:
3129:
3126:
3122:
3118:
3114:
3110:
3105:
3099:
3096:
3091:
3086:
3079:
3076:
3071:
3068:
3064:
3056:
3055:
3040:
3036:
3031:
3025:
3018:
3015:
3010:
3007:
3003:
2999:
2994:
2987:
2984:
2979:
2976:
2972:
2967:
2959:
2956:
2951:
2948:
2944:
2940:
2936:
2931:
2926:
2919:
2916:
2911:
2908:
2904:
2900:
2896:
2893:
2889:
2884:
2877:
2874:
2869:
2866:
2862:
2858:
2854:
2850:
2845:
2839:
2836:
2831:
2826:
2819:
2816:
2811:
2808:
2804:
2796:
2795:
2794:
2791:
2789:
2788:semi-discrete
2785:
2780:
2778:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2742:
2740:
2738:
2720:
2715:
2711:
2707:
2704:
2701:
2697:
2687:
2673:
2670:
2667:
2647:
2644:
2641:
2633:
2615:
2610:
2606:
2602:
2598:
2591:The function
2575:
2567:
2563:
2559:
2554:
2551:
2548:
2544:
2536:
2533:
2530:
2526:
2522:
2517:
2513:
2506:
2501:
2497:
2489:
2488:
2487:
2470:
2466:
2460:
2456:
2452:
2447:
2444:
2441:
2437:
2432:
2427:
2422:
2418:
2414:
2410:
2407:
2404:
2399:
2395:
2391:
2386:
2381:
2377:
2373:
2370:
2367:
2363:
2359:
2355:
2349:
2346:
2343:
2339:
2335:
2330:
2326:
2321:
2316:
2311:
2308:
2305:
2301:
2297:
2293:
2290:
2287:
2282:
2279:
2276:
2272:
2268:
2263:
2258:
2254:
2250:
2247:
2244:
2240:
2232:
2231:
2216:
2212:
2206:
2203:
2200:
2196:
2192:
2187:
2184:
2181:
2177:
2172:
2167:
2162:
2159:
2156:
2152:
2148:
2144:
2141:
2138:
2133:
2130:
2127:
2123:
2119:
2114:
2109:
2105:
2101:
2098:
2095:
2091:
2087:
2083:
2077:
2073:
2069:
2064:
2061:
2058:
2054:
2049:
2044:
2039:
2035:
2031:
2027:
2024:
2021:
2016:
2012:
2008:
2003:
1998:
1994:
1990:
1987:
1984:
1980:
1972:
1971:
1970:
1953:
1949:
1943:
1938:
1934:
1930:
1927:
1924:
1920:
1916:
1911:
1906:
1902:
1898:
1895:
1892:
1888:
1883:
1877:
1872:
1868:
1864:
1861:
1858:
1854:
1850:
1845:
1840:
1836:
1832:
1829:
1826:
1822:
1818:
1814:
1808:
1803:
1799:
1795:
1792:
1789:
1785:
1781:
1776:
1771:
1767:
1763:
1760:
1757:
1753:
1748:
1742:
1737:
1733:
1729:
1726:
1723:
1719:
1715:
1710:
1705:
1701:
1697:
1694:
1691:
1687:
1679:
1678:
1677:
1675:
1657:
1652:
1648:
1644:
1641:
1638:
1634:
1611:
1606:
1602:
1598:
1595:
1592:
1588:
1578:
1562:
1557:
1553:
1549:
1546:
1543:
1539:
1515:
1512:
1508:
1502:
1497:
1493:
1489:
1486:
1483:
1479:
1475:
1470:
1465:
1461:
1457:
1454:
1451:
1447:
1442:
1433:
1429:
1421:
1416:
1410:
1398:
1394:
1378:
1377:
1376:
1359:
1356:
1352:
1347:
1342:
1337:
1333:
1329:
1326:
1323:
1319:
1315:
1311:
1308:
1304:
1299:
1294:
1290:
1286:
1283:
1280:
1276:
1272:
1268:
1264:
1255:
1251:
1243:
1238:
1232:
1220:
1216:
1200:
1199:
1198:
1196:
1195:slope limited
1187:
1183:
1181:
1165:
1162:
1157:
1153:
1149:
1144:
1140:
1130:
1111:
1107:
1101:
1097:
1093:
1088:
1085:
1082:
1078:
1073:
1069:
1066:
1061:
1057:
1053:
1050:
1047:
1043:
1035:
1021:
1017:
1011:
1008:
1005:
1001:
997:
992:
988:
983:
979:
976:
971:
967:
963:
960:
957:
953:
945:
944:
943:
941:
923:
919:
915:
912:
909:
905:
882:
878:
874:
871:
868:
864:
839:
836:
831:
827:
823:
818:
814:
804:
786:
783:
780:
776:
771:
766:
762:
758:
755:
752:
748:
744:
740:
737:
733:
728:
724:
720:
717:
714:
710:
706:
702:
698:
689:
685:
677:
672:
666:
654:
650:
634:
633:
632:
615:
607:
604:
601:
597:
593:
588:
584:
577:
574:
566:
560:
556:
552:
547:
544:
541:
537:
532:
525:
519:
515:
511:
506:
503:
500:
496:
491:
486:
480:
476:
472:
469:
465:
459:
454:
450:
446:
442:
439:
436:
432:
425:
424:
423:
421:
417:
412:
410:
390:
387:
383:
378:
373:
370:
367:
363:
359:
355:
352:
348:
343:
339:
335:
331:
327:
318:
314:
306:
301:
295:
283:
279:
263:
262:
261:
247:
238:
236:
233:represents a
220:
200:
176:
173:
169:
166:
163:
157:
153:
149:
144:
140:
132:
131:
130:
112:
109:
104:
100:
96:
91:
87:
77:
70:
68:
66:
62:
61:
55:
50:
48:
47:
42:
38:
37:Bram van Leer
34:
30:
26:
22:
8883:Peridynamics
8776:
8701:Lax–Wendroff
8592:
8585:
8578:
8564:
8557:
8550:
8546:
8542:, pp267–279.
8539:
8535:
8525:
8521:
8510:
8506:
8499:
8490:
8481:
8477:
8468:
8464:
8461:Eitan Tadmor
8451:
8409:Flux limiter
8393:
8385:Liou-Steffen
8384:
8378:Osher scheme
8377:
8371:
8284:
8230:
8170:
8167:
8005:
8003:
7917:
7408:
7122:
7044:
6983:
6957:
6811:
6807:
6803:
6801:
6774:
6748:
6746:
6599:
6516:
6432:
6395:
6031:
5150:
4912:
4820:
4816:
4798:
4794:
4790:
4786:
4785:
4779:
4775:
4773:
4298:
4060:
3961:
3904:
3899:
3897:
3890:
3745:
3371:
3369:
2792:
2787:
2783:
2781:
2760:
2756:
2752:
2748:
2746:
2736:
2688:
2632:Flux limiter
2590:
2485:
1968:
1673:
1580:The symbols
1579:
1530:
1374:
1194:
1192:
1126:
939:
855:
630:
420:second-order
419:
415:
413:
405:
239:
192:
128:
64:
58:
51:
44:
40:
32:
25:MUSCL scheme
24:
18:
9017:Collocation
8528:, 408–463.
8471:, 241–282.
7961:. Velocity
6055:= 1/3 and,
3469:over cells
409:Runge–Kutta
9106:Categories
8706:MacCormack
8688:Hyperbolic
8553:, 785–794.
8513:, 101–136.
8445:References
8287:shock tube
8233:shock tube
6834:(density)
6747:and where
6521:reduce to
3370:Where the
237:variable.
41:high-order
9022:Level-set
9012:Multigrid
8962:Balancing
8664:Parabolic
8518:E. Tadmor
8303:ϕ
8205:∗
8190:±
8143:∗
8128:−
8115:−
8110:∗
8064:Δ
8055:−
7926:ρ
7889:ρ
7885:−
7870:ρ
7843:ϕ
7837:−
7828:ρ
7804:−
7797:ρ
7779:−
7772:ρ
7768:−
7759:ρ
7741:−
7726:ϕ
7712:−
7705:ρ
7681:−
7674:ρ
7636:ρ
7632:−
7617:ρ
7584:ϕ
7578:−
7563:ρ
7532:ρ
7514:−
7507:ρ
7503:−
7494:ρ
7467:ϕ
7452:ρ
7421:ρ
7369:−
7362:ρ
7338:−
7331:ρ
7320:∗
7305:−
7298:ρ
7289:∗
7274:−
7267:ρ
7230:ρ
7199:ρ
7188:∗
7166:ρ
7157:∗
7135:ρ
7053:γ
7016:−
7013:γ
7005:ρ
6930:ρ
6911:ρ
6822:ρ
6688:ρ
6672:ρ
6634:ρ
6627:ρ
6570:∂
6560:∂
6545:∂
6535:∂
6443:ϕ
6404:ϕ
6368:−
6357:−
6349:−
6318:−
6307:δ
6279:−
6229:δ
6193:−
6182:−
6149:−
6138:δ
6116:−
6066:δ
6040:κ
5943:−
5932:δ
5924:κ
5884:δ
5876:κ
5873:−
5833:ϕ
5827:−
5794:−
5743:−
5732:δ
5724:κ
5695:−
5684:δ
5676:κ
5673:−
5642:−
5627:ϕ
5613:−
5582:−
5520:δ
5512:κ
5472:δ
5464:κ
5461:−
5415:ϕ
5409:−
5308:δ
5300:κ
5271:−
5260:δ
5252:κ
5249:−
5209:ϕ
5111:−
5080:−
5056:∗
5041:−
4949:∗
4897:∗
4882:−
4852:∗
4738:−
4727:Δ
4717:−
4706:−
4653:−
4642:Δ
4632:−
4621:−
4597:−
4548:−
4491:Δ
4476:−
4406:Δ
4391:−
4264:−
4253:−
4209:Δ
4190:∗
4175:−
4164:−
4159:∗
4115:Δ
4106:−
3868:∂
3836:∂
3796:∂
3764:∂
3754:ρ
3713:∂
3655:∂
3645:ρ
3629:∂
3571:∂
3561:ρ
3505:given by
3489:±
3390:±
3260:−
3201:−
3087:∗
3011:−
3000:−
2980:−
2952:−
2941:−
2912:−
2870:−
2827:∗
2812:−
2721:∗
2645:≤
2599:ϕ
2560:−
2534:−
2523:−
2453:−
2411:ϕ
2405:−
2371:−
2347:−
2336:−
2309:−
2294:ϕ
2280:−
2248:−
2193:−
2145:ϕ
2139:−
2070:−
2028:ϕ
1928:−
1896:−
1878:∗
1862:−
1846:∗
1830:−
1743:∗
1711:∗
1658:∗
1642:−
1612:∗
1563:∗
1547:±
1503:∗
1487:−
1476:−
1471:∗
1426:Δ
1343:∗
1327:−
1309:−
1300:∗
1248:Δ
1086:−
1051:−
913:−
756:−
738:−
682:Δ
578:∈
572:∀
553:−
512:−
473:−
371:−
353:−
311:Δ
8996:Spectral
8935:additive
8858:Smoothed
8824:Extended
8398:See also
8281:limiter.
8227:limiter.
6808:momentum
8980:FETI-DP
8860:(S-FEM)
8779:(MUSCL)
8767:Godunov
8610:Fortran
4299:Where,
8989:Others
8976:(FETI)
8970:(BDDC)
8842:Mortar
8826:(XFEM)
8819:hp-FEM
8802:(WENO)
8785:(AUSM)
8746:(FDTD)
8740:(FDFD)
8725:Others
8711:Upwind
8674:(FTCS)
8357:
7409:where
7056:
7045:where
6969:
6958:where
6812:energy
6810:, and
6600:where
6497:
6418:
6043:
6032:Where
3942:
3809:
3405:
2773:vector
2769:scalar
856:where
193:Where
23:, the
9003:(DVR)
8964:(BDD)
8903:(PIC)
8897:(MPM)
8891:(MPS)
8879:(SPH)
8849:(GDM)
8838:(SEM)
8796:(ENO)
8734:(ADI)
8380:, and
4787:Note:
2751:(KT)
27:is a
8885:(PD)
8832:(DG)
8606:GEES
8388:AUSM
8383:the
8376:the
8006:i.e.
6804:mass
5151:and
4913:i.e.
4866:and
4797:and
4778:and
4776:full
3825:of
3746:and
2771:and
2486:and
1674:i.e.
1626:and
940:i.e.
897:and
235:flux
8509:.,
8469:160
7840:0.5
7723:0.5
7581:0.5
7464:0.5
3893:CFL
3553:max
2786:or
2408:0.5
2291:0.5
2142:0.5
2025:0.5
1070:0.5
980:0.5
9108::
8551:83
8549:,
8538:,
8526:87
8524:,
8511:32
8482:22
8480:,
8467:,
8242:SI
8219:.
8008:,
6806:,
4915:,
3374:,
1676:,
1516:0.
1360:0.
942:,
391:0.
177:0.
43:,
8639:e
8632:t
8625:v
8540:1
8531:.
8349:2
8345:r
8341:+
8338:1
8333:r
8330:2
8324:=
8321:)
8318:r
8315:(
8310:a
8307:v
8198:2
8195:1
8187:i
8182:F
8153:.
8149:]
8136:2
8133:1
8125:i
8120:F
8103:2
8100:1
8095:+
8092:i
8087:F
8081:[
8072:i
8068:x
8060:1
8052:=
8046:t
8042:d
8034:i
8029:U
8023:d
7989:p
7969:u
7949:E
7929:u
7903:.
7899:)
7893:i
7880:1
7877:+
7874:i
7865:(
7860:)
7855:i
7851:r
7847:(
7832:i
7824:=
7819:R
7812:2
7809:1
7801:i
7792:,
7788:)
7782:1
7776:i
7763:i
7754:(
7749:)
7744:1
7738:i
7734:r
7730:(
7720:+
7715:1
7709:i
7701:=
7696:L
7689:2
7686:1
7678:i
7650:,
7646:)
7640:i
7627:1
7624:+
7621:i
7612:(
7607:)
7602:1
7599:+
7596:i
7592:r
7588:(
7573:1
7570:+
7567:i
7559:=
7554:R
7547:2
7544:1
7539:+
7536:i
7527:,
7523:)
7517:1
7511:i
7498:i
7489:(
7484:)
7479:i
7475:r
7471:(
7461:+
7456:i
7448:=
7443:L
7436:2
7433:1
7428:+
7425:i
7394:,
7390:)
7384:R
7377:2
7374:1
7366:i
7358:,
7353:L
7346:2
7343:1
7335:i
7326:(
7313:2
7310:1
7302:i
7294:=
7282:2
7279:1
7271:i
7262:,
7258:)
7252:R
7245:2
7242:1
7237:+
7234:i
7226:,
7221:L
7214:2
7211:1
7206:+
7203:i
7194:(
7181:2
7178:1
7173:+
7170:i
7162:=
7150:2
7147:1
7142:+
7139:i
7107:]
7101:v
7097:c
7092:/
7086:p
7082:c
7077:[
7030:,
7027:e
7023:)
7019:1
7009:(
7002:=
6999:p
6966:e
6943:,
6938:2
6934:u
6925:2
6922:1
6917:+
6914:e
6908:=
6905:E
6882:E
6862:p
6842:u
6784:F
6758:U
6731:,
6726:)
6720:)
6717:p
6714:+
6711:E
6708:(
6705:u
6696:2
6692:u
6685:+
6682:p
6675:u
6666:(
6661:=
6657:F
6650:)
6644:E
6637:u
6621:(
6616:=
6612:U
6585:,
6582:0
6579:=
6573:x
6564:F
6554:+
6548:t
6539:U
6489:2
6485:r
6481:+
6478:1
6473:r
6470:2
6464:=
6461:)
6458:r
6455:(
6450:a
6447:v
6414:)
6411:r
6408:(
6381:,
6377:)
6371:2
6365:i
6361:u
6352:1
6346:i
6342:u
6337:(
6333:=
6326:2
6323:3
6315:i
6311:u
6303:,
6299:)
6293:1
6290:+
6287:i
6283:u
6274:2
6271:+
6268:i
6264:u
6259:(
6255:=
6248:2
6245:3
6240:+
6237:i
6233:u
6206:,
6202:)
6196:1
6190:i
6186:u
6177:i
6173:u
6168:(
6164:=
6157:2
6154:1
6146:i
6142:u
6134:,
6130:)
6124:i
6120:u
6111:1
6108:+
6105:i
6101:u
6096:(
6092:=
6085:2
6082:1
6077:+
6074:i
6070:u
6016:0
6013:=
6008:x
6004:u
6000:+
5995:t
5991:u
5963:.
5959:]
5951:2
5948:1
5940:i
5936:u
5928:)
5921:+
5918:1
5914:(
5910:+
5903:2
5900:1
5895:+
5892:i
5888:u
5880:)
5870:1
5866:(
5861:[
5855:4
5850:)
5845:i
5841:r
5837:(
5822:i
5818:u
5814:=
5809:R
5802:2
5799:1
5791:i
5787:u
5763:,
5759:]
5751:2
5748:1
5740:i
5736:u
5728:)
5721:+
5718:1
5714:(
5710:+
5703:2
5700:3
5692:i
5688:u
5680:)
5670:1
5666:(
5661:[
5655:4
5650:)
5645:1
5639:i
5635:r
5631:(
5621:+
5616:1
5610:i
5606:u
5602:=
5597:L
5590:2
5587:1
5579:i
5575:u
5551:,
5547:]
5539:2
5536:1
5531:+
5528:i
5524:u
5516:)
5509:+
5506:1
5502:(
5498:+
5491:2
5488:3
5483:+
5480:i
5476:u
5468:)
5458:1
5454:(
5449:[
5443:4
5438:)
5433:1
5430:+
5427:i
5423:r
5419:(
5404:1
5401:+
5398:i
5394:u
5390:=
5385:R
5378:2
5375:1
5370:+
5367:i
5363:u
5339:,
5335:]
5327:2
5324:1
5319:+
5316:i
5312:u
5304:)
5297:+
5294:1
5290:(
5286:+
5279:2
5276:1
5268:i
5264:u
5256:)
5246:1
5242:(
5237:[
5231:4
5226:)
5221:i
5217:r
5213:(
5203:+
5198:i
5194:u
5190:=
5185:L
5178:2
5175:1
5170:+
5167:i
5163:u
5136:,
5132:)
5126:R
5119:2
5116:1
5108:i
5104:u
5100:,
5095:L
5088:2
5085:1
5077:i
5073:u
5068:(
5064:f
5061:=
5049:2
5046:1
5038:i
5034:u
5029:,
5025:)
5019:R
5012:2
5009:1
5004:+
5001:i
4997:u
4993:,
4988:L
4981:2
4978:1
4973:+
4970:i
4966:u
4961:(
4957:f
4954:=
4942:2
4939:1
4934:+
4931:i
4927:u
4890:2
4887:1
4879:i
4875:u
4845:2
4842:1
4837:+
4834:i
4830:u
4758:]
4754:.
4750:)
4741:1
4735:i
4731:x
4720:1
4714:i
4710:u
4701:i
4697:u
4690:,
4685:i
4681:u
4676:(
4672:Q
4669:+
4665:)
4656:1
4650:i
4646:x
4635:1
4629:i
4625:u
4616:i
4612:u
4605:,
4600:1
4594:i
4590:u
4585:(
4581:Q
4577:[
4571:2
4568:1
4563:=
4556:2
4553:1
4545:i
4541:P
4517:,
4513:]
4508:)
4499:i
4495:x
4484:i
4480:u
4471:1
4468:+
4465:i
4461:u
4454:,
4449:1
4446:+
4443:i
4439:u
4434:(
4430:Q
4427:+
4423:)
4414:i
4410:x
4399:i
4395:u
4386:1
4383:+
4380:i
4376:u
4369:,
4364:i
4360:u
4355:(
4351:Q
4347:[
4341:2
4338:1
4333:=
4326:2
4323:1
4318:+
4315:i
4311:P
4284:.
4280:]
4272:2
4269:1
4261:i
4257:P
4246:2
4243:1
4238:+
4235:i
4231:P
4226:[
4217:i
4213:x
4205:1
4200:+
4196:]
4183:2
4180:1
4172:i
4168:F
4152:2
4149:1
4144:+
4141:i
4137:F
4132:[
4123:i
4119:x
4111:1
4103:=
4097:t
4093:d
4085:i
4081:u
4076:d
4046:,
4042:)
4036:x
4032:u
4028:,
4025:u
4021:(
4015:x
4011:Q
4007:=
4003:)
4000:u
3997:(
3991:x
3987:F
3983:+
3978:t
3974:u
3939:0
3936:=
3931:x
3927:u
3923:+
3918:t
3914:u
3877:.
3871:u
3862:)
3857:)
3854:t
3851:(
3847:u
3843:(
3839:F
3805:)
3799:u
3790:)
3785:)
3782:t
3779:(
3775:u
3771:(
3767:F
3758:(
3730:]
3726:,
3722:)
3716:u
3707:)
3702:)
3699:t
3696:(
3690:R
3685:2
3681:/
3677:1
3674:+
3671:i
3667:u
3662:(
3658:F
3649:(
3642:,
3638:)
3632:u
3623:)
3618:)
3615:t
3612:(
3606:L
3601:2
3597:/
3593:1
3590:+
3587:i
3583:u
3578:(
3574:F
3565:(
3557:[
3550:=
3546:)
3543:t
3540:(
3532:2
3529:1
3524:+
3521:i
3517:a
3492:1
3486:i
3482:,
3478:i
3456:)
3451:)
3447:t
3444:,
3441:x
3437:(
3433:u
3429:(
3425:F
3398:2
3395:1
3387:i
3383:a
3354:0
3351:=
3346:x
3342:u
3338:+
3333:t
3329:u
3301:.
3297:}
3292:]
3286:L
3279:2
3276:1
3271:+
3268:i
3264:u
3255:R
3248:2
3245:1
3240:+
3237:i
3233:u
3228:[
3220:2
3217:1
3212:+
3209:i
3205:a
3197:]
3192:)
3187:L
3180:2
3177:1
3172:+
3169:i
3165:u
3161:(
3157:F
3154:+
3150:)
3145:R
3138:2
3135:1
3130:+
3127:i
3123:u
3119:(
3115:F
3111:[
3106:{
3100:2
3097:1
3092:=
3080:2
3077:1
3072:+
3069:i
3065:F
3041:.
3037:}
3032:]
3026:L
3019:2
3016:1
3008:i
3004:u
2995:R
2988:2
2985:1
2977:i
2973:u
2968:[
2960:2
2957:1
2949:i
2945:a
2937:]
2932:)
2927:L
2920:2
2917:1
2909:i
2905:u
2901:(
2897:F
2894:+
2890:)
2885:R
2878:2
2875:1
2867:i
2863:u
2859:(
2855:F
2851:[
2846:{
2840:2
2837:1
2832:=
2820:2
2817:1
2809:i
2805:F
2716:2
2712:/
2708:1
2705:+
2702:i
2698:F
2674:1
2671:=
2668:r
2648:0
2642:r
2616:)
2611:i
2607:r
2603:(
2576:.
2568:i
2564:u
2555:1
2552:+
2549:i
2545:u
2537:1
2531:i
2527:u
2518:i
2514:u
2507:=
2502:i
2498:r
2471:,
2467:)
2461:i
2457:u
2448:1
2445:+
2442:i
2438:u
2433:(
2428:)
2423:i
2419:r
2415:(
2400:i
2396:u
2392:=
2387:R
2382:2
2378:/
2374:1
2368:i
2364:u
2360:,
2356:)
2350:1
2344:i
2340:u
2331:i
2327:u
2322:(
2317:)
2312:1
2306:i
2302:r
2298:(
2288:+
2283:1
2277:i
2273:u
2269:=
2264:L
2259:2
2255:/
2251:1
2245:i
2241:u
2217:,
2213:)
2207:1
2204:+
2201:i
2197:u
2188:2
2185:+
2182:i
2178:u
2173:(
2168:)
2163:1
2160:+
2157:i
2153:r
2149:(
2134:1
2131:+
2128:i
2124:u
2120:=
2115:R
2110:2
2106:/
2102:1
2099:+
2096:i
2092:u
2088:,
2084:)
2078:i
2074:u
2065:1
2062:+
2059:i
2055:u
2050:(
2045:)
2040:i
2036:r
2032:(
2022:+
2017:i
2013:u
2009:=
2004:L
1999:2
1995:/
1991:1
1988:+
1985:i
1981:u
1954:,
1950:)
1944:R
1939:2
1935:/
1931:1
1925:i
1921:u
1917:,
1912:L
1907:2
1903:/
1899:1
1893:i
1889:u
1884:(
1873:2
1869:/
1865:1
1859:i
1855:u
1851:=
1841:2
1837:/
1833:1
1827:i
1823:u
1819:,
1815:)
1809:R
1804:2
1800:/
1796:1
1793:+
1790:i
1786:u
1782:,
1777:L
1772:2
1768:/
1764:1
1761:+
1758:i
1754:u
1749:(
1738:2
1734:/
1730:1
1727:+
1724:i
1720:u
1716:=
1706:2
1702:/
1698:1
1695:+
1692:i
1688:u
1653:2
1649:/
1645:1
1639:i
1635:u
1607:2
1603:/
1599:1
1596:+
1593:i
1589:u
1558:2
1554:/
1550:1
1544:i
1540:F
1513:=
1509:]
1498:2
1494:/
1490:1
1484:i
1480:F
1466:2
1462:/
1458:1
1455:+
1452:i
1448:F
1443:[
1434:i
1430:x
1422:1
1417:+
1411:t
1407:d
1399:i
1395:u
1390:d
1357:=
1353:]
1348:)
1338:2
1334:/
1330:1
1324:i
1320:u
1316:(
1312:F
1305:)
1295:2
1291:/
1287:1
1284:+
1281:i
1277:u
1273:(
1269:F
1265:[
1256:i
1252:x
1244:1
1239:+
1233:t
1229:d
1221:i
1217:u
1212:d
1166:0
1163:=
1158:x
1154:u
1150:+
1145:t
1141:u
1112:.
1108:)
1102:i
1098:u
1094:+
1089:1
1083:i
1079:u
1074:(
1067:=
1062:2
1058:/
1054:1
1048:i
1044:u
1022:,
1018:)
1012:1
1009:+
1006:i
1002:u
998:+
993:i
989:u
984:(
977:=
972:2
968:/
964:1
961:+
958:i
954:u
924:2
920:/
916:1
910:i
906:u
883:2
879:/
875:1
872:+
869:i
865:u
840:0
837:=
832:x
828:u
824:+
819:t
815:u
787:,
784:0
781:=
777:]
772:)
767:2
763:/
759:1
753:i
749:u
745:(
741:F
734:)
729:2
725:/
721:1
718:+
715:i
711:u
707:(
703:F
699:[
690:i
686:x
678:1
673:+
667:t
663:d
655:i
651:u
646:d
616:.
613:]
608:1
605:+
602:i
598:x
594:,
589:i
585:x
581:(
575:x
567:)
561:i
557:u
548:1
545:+
542:i
538:u
533:(
526:)
520:i
516:x
507:1
504:+
501:i
497:x
492:(
487:)
481:i
477:x
470:x
466:(
460:+
455:i
451:u
447:=
443:)
440:x
437:(
433:u
388:=
384:]
379:)
374:1
368:i
364:u
360:(
356:F
349:)
344:i
340:u
336:(
332:F
328:[
319:i
315:x
307:1
302:+
296:t
292:d
284:i
280:u
275:d
248:i
221:F
201:u
174:=
170:)
167:u
164:(
158:x
154:F
150:+
145:t
141:u
113:0
110:=
105:x
101:u
97:+
92:t
88:u
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