1393:
The above shows how to use the method to propagate a solution forward in space; however, many physics applications, such as studying the evolution of a wave packet describing a particle, require one to propagate the solution forward in time rather than in space. The non-linear Schrödinger equation,
1573:
59:
An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem. Another
2858:
A variation on this method is the symmetrized split-step
Fourier method, which takes half a time step using one operator, then takes a full-time step with only the other, and then takes a second half time step again with only the first. This method is an improvement upon the generic split-step
2383:
is the frequency (or more properly, wave number, as we are dealing with a spatial variable and thus transforming to a space of spatial frequencies—i.e. wave numbers) associated with the
Fourier transform of whatever is being operated on. Thus, we take the Fourier transform of
261:
469:
2226:
1742:
1025:
1400:
1223:
572:
1961:
44:. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). Second, it is necessary to
93:
2459:
347:
757:
1797:
2361:
817:
1280:
845:
2852:
1325:
3246:
2950:; Sylvestre, Thibaut; Randle, Hamish G.; Coen, Stéphane (2013-01-01). "Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model".
2675:
2509:
1056:
2571:
2542:
2308:
2259:
2023:
1994:
1611:
2088:
2887:
2057:
1837:
299:
879:
3147:
T. R. Taha and M. J. Ablowitz (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation".
2910:
2331:
2080:
1388:
2699:
2381:
2279:
1817:
1651:
1631:
1365:
1345:
622:
602:
339:
319:
1659:
887:
1568:{\displaystyle i\hbar {\partial \psi \over \partial t}=-{{\hbar }^{2} \over {2m}}{\partial ^{2}\psi \over \partial x^{2}}+\gamma |\psi |^{2}\psi =\psi ,}
2701:
denotes a
Fourier transform. We then inverse Fourier transform this expression to find the final result in physical space, yielding the final expression
2026:
3711:
3337:
1068:
480:
3287:
3263:
3237:
1853:
3561:
3331:
3631:
3488:
3343:
624:, then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,
3019:
Maleki, L.; Seidel, D.; Ilchenko, V. S.; Liang, W.; Savchenkov, A. A.; Matsko, A. B. (2011-08-01). "Mode-locked Kerr frequency combs".
3131:
578:
85:
41:
3556:
3539:
3690:
3476:
3457:
3446:
3423:
256:{\displaystyle {\partial A \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}+i\gamma |A|^{2}A=A,}
3429:
464:{\displaystyle {\partial A_{D} \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}={\hat {D}}A,}
37:
3546:
2390:
69:
3511:
2310:
we use the fact that in frequency space, the partial derivative operator can be converted into a number by substituting
3551:
3198:
3230:
3668:
1062:, the analytical solution to the linear step, commuted with the frequency domain solution for the nonlinear step, is
630:
3653:
3529:
1747:
3295:
3277:
60:
application of the split-step method that has been gaining a lot of traction since the 2010s is the simulation of
3315:
1229:
72:
with reasonable numerical cost, along with its success in reproducing experimental spectra as well as predicting
2336:
765:
3638:
3524:
3254:
2927:
1235:
3300:
822:
3680:
3658:
3643:
3626:
3534:
3519:
3435:
1840:
65:
2707:
3600:
3371:
3223:
2922:
33:
1285:
3716:
3648:
3494:
3410:
3215:
3071:
3685:
3358:
3156:
3086:
3028:
2969:
3452:
3366:
2579:
61:
2471:
3675:
3616:
3001:
2959:
2221:{\displaystyle \psi (x,t+dt)\approx e^{-idt{\hat {D}}/\hbar }e^{-idt{\hat {N}}/\hbar }\psi (x,t)}
1034:
17:
2547:
2518:
2284:
2235:
1999:
1970:
1581:
3305:
3127:
3104:
3052:
3044:
2993:
2985:
2913:
1059:
45:
1058:
is the center frequency of the pulse. It can be shown that using the above definition of the
3621:
3611:
3500:
3468:
3210:
3164:
3094:
3036:
2977:
2862:
2032:
1822:
851:
269:
49:
857:
3663:
3606:
3595:
1737:{\displaystyle {\hat {D}}=-{{\hbar }^{2} \over {2m}}{\partial ^{2} \over \partial x^{2}}}
3160:
3090:
3032:
2973:
2892:
2313:
2062:
2029:
can be applied to show that the error from treating them as if they do will be of order
1370:
3441:
3388:
2947:
2684:
2366:
2264:
1802:
1636:
1616:
1350:
1330:
607:
587:
324:
304:
1020:{\displaystyle {\tilde {A}}_{N}(\omega ,z)=\int _{-\infty }^{\infty }A_{N}(t,z)\expdt}
3705:
3310:
3168:
3199:
http://www.optics.rochester.edu/workgroups/agrawal/grouphomepage.php?pageid=software
3482:
3399:
3376:
3005:
3393:
3271:
3186:
1218:{\displaystyle {\tilde {A}}(\omega ,z+h)=\exp \left{\tilde {A}}_{N}(\omega ,z).}
53:
1847:
The formal solution to this equation is a complex exponential, so we have that
567:{\displaystyle {\partial A_{N} \over \partial z}=i\gamma |A|^{2}A={\hat {N}}A.}
3108:
3099:
3048:
2989:
3072:"Dynamics of microresonator frequency comb generation: models and stability"
2917:
3056:
2997:
577:
Both the linear and the nonlinear parts have analytical solutions, but the
2926:. The split-step Fourier method can therefore be much faster than typical
1956:{\displaystyle \psi (x,t)=e^{-it({\hat {D}}+{\hat {N}})/\hbar }\psi (x,0)}
1394:
when used to govern the time evolution of a wave function, takes the form
3040:
2981:
3579:
73:
3418:
2515:
and use it to find the product of the complex exponentials involving
76:
behavior in these microresonators has made the method very popular.
3192:
581:
containing both parts does not have a general analytical solution.
2964:
3573:
3567:
3382:
3219:
762:
using the analytical solution. Note that this ansatz imposes
2025:
are operators, they do not in general commute. However, the
3204:
3211:
http://www.mathworks.com/matlabcentral/fileexchange/24016
2465:
recover the associated wave number, compute the quantity
2261:
can be computed directly using the wave function at time
850:
The dispersion step has an analytical solution in the
48:
back and forth because the linear step is made in the
2895:
2865:
2710:
2687:
2582:
2550:
2521:
2474:
2393:
2369:
2340:
2316:
2287:
2267:
2238:
2091:
2065:
2035:
2002:
1973:
1856:
1825:
1805:
1750:
1662:
1639:
1619:
1584:
1403:
1373:
1353:
1333:
1288:
1238:
1071:
1037:
890:
860:
825:
768:
633:
610:
590:
483:
350:
327:
307:
272:
96:
3247:
Numerical methods for partial differential equations
3126:(3rd ed.). San Diego, CA, USA: Academic Press.
1367:
times, the pulse can be propagated over a length of
3588:
3510:
3467:
3409:
3357:
3324:
3286:
3262:
3253:
2454:{\displaystyle e^{-idt{\hat {N}}/\hbar }\psi (x,t)}
2904:
2881:
2846:
2693:
2669:
2565:
2536:
2503:
2453:
2375:
2355:
2325:
2302:
2273:
2253:
2220:
2074:
2051:
2017:
1988:
1955:
1831:
1811:
1791:
1736:
1645:
1625:
1605:
1567:
1382:
1359:
1339:
1327:; the pulse has thus been propagated a small step
1319:
1274:
1217:
1050:
1019:
873:
839:
811:
751:
616:
596:
566:
463:
333:
313:
293:
255:
854:, so it is first necessary to Fourier transform
341:. The equation can be split into a linear part,
2059:if we are taking a small but finite time step
752:{\displaystyle A_{N}(t,z+h)=\exp \leftA(t,z),}
3231:
3187:http://www.photonics.umd.edu/software/ssprop/
2859:Fourier method because its error is of order
1792:{\displaystyle {\hat {N}}=\gamma |\psi |^{2}}
68:. The relative ease of implementation of the
8:
2281:, but to compute the exponential involving
3259:
3238:
3224:
3216:
2920:can be computed relatively fast using the
2356:{\displaystyle \partial \over \partial x}
812:{\displaystyle |A(z)|^{2}={\text{const}}.}
3098:
3070:Hansson, Tobias; Wabnitz, Stefan (2016).
2963:
2894:
2873:
2864:
2807:
2806:
2793:
2775:
2761:
2745:
2709:
2686:
2633:
2632:
2619:
2601:
2587:
2581:
2552:
2551:
2549:
2523:
2522:
2520:
2493:
2479:
2473:
2423:
2412:
2411:
2398:
2392:
2368:
2338:
2315:
2289:
2288:
2286:
2266:
2240:
2239:
2237:
2190:
2179:
2178:
2165:
2151:
2140:
2139:
2126:
2090:
2064:
2043:
2034:
2004:
2003:
2001:
1975:
1974:
1972:
1925:
1911:
1910:
1896:
1895:
1882:
1855:
1824:
1804:
1783:
1778:
1769:
1752:
1751:
1749:
1725:
1711:
1705:
1695:
1689:
1684:
1681:
1664:
1663:
1661:
1638:
1618:
1583:
1545:
1544:
1530:
1529:
1514:
1509:
1500:
1485:
1467:
1460:
1450:
1444:
1439:
1436:
1410:
1402:
1372:
1352:
1332:
1287:
1275:{\displaystyle {\tilde {A}}(\omega ,z+h)}
1240:
1239:
1237:
1191:
1180:
1179:
1164:
1154:
1129:
1119:
1073:
1072:
1070:
1042:
1036:
996:
950:
940:
932:
904:
893:
892:
889:
865:
859:
833:
832:
824:
801:
792:
787:
769:
767:
714:
709:
685:
638:
632:
609:
589:
547:
546:
534:
529:
520:
494:
484:
482:
444:
443:
431:
413:
406:
394:
384:
361:
351:
349:
326:
306:
271:
233:
232:
218:
217:
202:
197:
188:
170:
152:
145:
133:
123:
97:
95:
36:numerical method used to solve nonlinear
1613:describes the wave function at position
52:while the nonlinear step is made in the
2939:
2428:
2195:
2156:
1930:
1826:
1685:
1440:
1407:
840:{\displaystyle \gamma \in \mathbb {R} }
2847:{\displaystyle \psi (x,t+dt)=F^{-1}]}
301:describes the pulse envelope in time
7:
3489:Moving particle semi-implicit method
3400:Weighted essentially non-oscillatory
2232:The part of this equation involving
1320:{\displaystyle A\left(t,z+h\right)}
3338:Finite-difference frequency-domain
2346:
2341:
1718:
1708:
1478:
1464:
1421:
1413:
941:
936:
502:
487:
424:
410:
369:
354:
163:
149:
108:
100:
14:
3712:Numerical differential equations
3193:http://www.freeopticsproject.org
2027:Baker-Campbell-Hausdorff formula
1819:is the mass of the particle and
584:However, if only a 'small' step
3691:Method of fundamental solutions
3477:Smoothed-particle hydrodynamics
3332:Alternating direction-implicit
2841:
2838:
2835:
2823:
2812:
2786:
2754:
2735:
2714:
2664:
2661:
2649:
2638:
2612:
2557:
2528:
2448:
2436:
2417:
2294:
2245:
2215:
2203:
2184:
2145:
2116:
2095:
2009:
1980:
1950:
1938:
1922:
1916:
1901:
1892:
1872:
1860:
1779:
1770:
1757:
1669:
1600:
1588:
1556:
1550:
1535:
1526:
1510:
1501:
1269:
1251:
1245:
1209:
1197:
1185:
1161:
1141:
1102:
1084:
1078:
1008:
1002:
983:
977:
968:
956:
922:
910:
898:
788:
783:
777:
770:
743:
731:
710:
705:
693:
686:
662:
644:
579:nonlinear Schrödinger equation
552:
530:
521:
449:
288:
276:
244:
238:
223:
214:
198:
189:
86:nonlinear Schrödinger equation
42:nonlinear Schrödinger equation
38:partial differential equations
1:
3344:Finite-difference time-domain
3191:Andrés A. Rieznik, Software,
2670:{\displaystyle e^{idtk^{2}}F}
2573:in frequency space as below:
3383:Advection upstream-splitting
3203:Thomas Schreiber, Software,
3197:Prof. G. Agrawal, Software,
3185:Thomas E. Murphy, Software,
3169:10.1016/0021-9991(84)90003-2
2923:fast Fourier transform (FFT)
2504:{\displaystyle e^{idtk^{2}}}
3394:Essentially non-oscillatory
3377:Monotonic upstream-centered
3209:Edward J. Grace, Software,
3122:Agrawal, Govind P. (2001).
1051:{\displaystyle \omega _{0}}
84:Consider, for example, the
3733:
3654:Infinite difference method
3272:Forward-time central-space
2566:{\displaystyle {\hat {D}}}
2537:{\displaystyle {\hat {N}}}
2303:{\displaystyle {\hat {D}}}
2254:{\displaystyle {\hat {N}}}
2018:{\displaystyle {\hat {N}}}
1989:{\displaystyle {\hat {D}}}
1606:{\displaystyle \psi (x,t)}
3557:Poincaré–Steklov operator
3316:Method of characteristics
2928:finite difference methods
2082:. We therefore can write
1347:. By repeating the above
1230:inverse Fourier transform
80:Description of the method
3574:Tearing and interconnect
3568:Balancing by constraints
3205:http://www.fiberdesk.com
3100:10.1515/nanoph-2016-0012
321:at the spatial position
70:Lugiato–Lefever equation
3681:Computer-assisted proof
3659:Infinite element method
3447:Gradient discretisation
1841:reduced Planck constant
66:optical microresonators
3669:Petrov–Galerkin method
3430:Discontinuous Galerkin
3124:Nonlinear Fiber Optics
2906:
2883:
2882:{\displaystyle dt^{3}}
2848:
2695:
2671:
2567:
2538:
2505:
2455:
2377:
2357:
2327:
2304:
2275:
2255:
2222:
2076:
2053:
2052:{\displaystyle dt^{2}}
2019:
1990:
1957:
1833:
1832:{\displaystyle \hbar }
1813:
1793:
1738:
1647:
1627:
1607:
1569:
1384:
1361:
1341:
1321:
1276:
1219:
1052:
1021:
875:
841:
813:
753:
618:
598:
568:
474:and a nonlinear part,
465:
335:
315:
295:
294:{\displaystyle A(t,z)}
257:
3649:Isogeometric analysis
3495:Material point method
2907:
2884:
2849:
2696:
2672:
2568:
2539:
2506:
2456:
2378:
2358:
2328:
2305:
2276:
2256:
2223:
2077:
2054:
2020:
1991:
1958:
1834:
1814:
1794:
1739:
1648:
1628:
1608:
1570:
1385:
1362:
1342:
1322:
1277:
1220:
1053:
1022:
876:
874:{\displaystyle A_{N}}
842:
814:
754:
619:
599:
569:
466:
336:
316:
296:
258:
3686:Integrable algorithm
3512:Domain decomposition
3041:10.1364/OL.36.002845
2982:10.1364/OL.38.000037
2893:
2863:
2708:
2685:
2580:
2548:
2519:
2472:
2391:
2367:
2337:
2314:
2285:
2265:
2236:
2089:
2063:
2033:
2000:
1971:
1854:
1823:
1803:
1748:
1660:
1637:
1617:
1582:
1401:
1371:
1351:
1331:
1286:
1236:
1069:
1035:
888:
858:
823:
766:
631:
608:
588:
481:
348:
325:
305:
270:
94:
3530:Schwarz alternating
3453:Loubignac iteration
3180:External references
3161:1984JCoPh..55..203T
3091:2016Nanop...5...12H
3033:2011OptL...36.2845M
2974:2013OptL...38...37C
945:
62:Kerr frequency comb
3676:Validated numerics
2914:Fourier transforms
2905:{\displaystyle dt}
2902:
2879:
2844:
2691:
2667:
2563:
2534:
2501:
2451:
2373:
2344:
2326:{\displaystyle ik}
2323:
2300:
2271:
2251:
2218:
2075:{\displaystyle dt}
2072:
2049:
2015:
1986:
1953:
1829:
1809:
1789:
1734:
1643:
1623:
1603:
1565:
1383:{\displaystyle Nh}
1380:
1357:
1337:
1317:
1272:
1215:
1048:
1017:
928:
871:
837:
809:
749:
614:
594:
564:
461:
331:
311:
291:
253:
18:numerical analysis
3699:
3698:
3639:Immersed boundary
3632:Method of moments
3547:Neumann–Dirichlet
3540:abstract additive
3525:Fictitious domain
3469:Meshless/Meshfree
3353:
3352:
3255:Finite difference
3027:(15): 2845–2847.
2815:
2694:{\displaystyle F}
2641:
2560:
2531:
2420:
2376:{\displaystyle k}
2353:
2297:
2274:{\displaystyle t}
2248:
2187:
2148:
2012:
1983:
1919:
1904:
1812:{\displaystyle m}
1760:
1732:
1703:
1672:
1646:{\displaystyle t}
1626:{\displaystyle x}
1553:
1538:
1492:
1458:
1428:
1360:{\displaystyle N}
1340:{\displaystyle h}
1248:
1188:
1139:
1081:
1060:Fourier transform
901:
819:and consequently
804:
617:{\displaystyle z}
597:{\displaystyle h}
555:
509:
452:
438:
404:
376:
334:{\displaystyle z}
314:{\displaystyle t}
241:
226:
177:
143:
115:
46:Fourier transform
3724:
3644:Analytic element
3627:Boundary element
3520:Schur complement
3501:Particle-in-cell
3436:Spectral element
3260:
3240:
3233:
3226:
3217:
3173:
3172:
3144:
3138:
3137:
3119:
3113:
3112:
3102:
3076:
3067:
3061:
3060:
3016:
3010:
3009:
2967:
2944:
2911:
2909:
2908:
2903:
2889:for a time step
2888:
2886:
2885:
2880:
2878:
2877:
2853:
2851:
2850:
2845:
2819:
2818:
2817:
2816:
2808:
2782:
2781:
2780:
2779:
2753:
2752:
2700:
2698:
2697:
2692:
2676:
2674:
2673:
2668:
2645:
2644:
2643:
2642:
2634:
2608:
2607:
2606:
2605:
2572:
2570:
2569:
2564:
2562:
2561:
2553:
2543:
2541:
2540:
2535:
2533:
2532:
2524:
2510:
2508:
2507:
2502:
2500:
2499:
2498:
2497:
2460:
2458:
2457:
2452:
2432:
2431:
2427:
2422:
2421:
2413:
2382:
2380:
2379:
2374:
2362:
2360:
2359:
2354:
2352:
2339:
2332:
2330:
2329:
2324:
2309:
2307:
2306:
2301:
2299:
2298:
2290:
2280:
2278:
2277:
2272:
2260:
2258:
2257:
2252:
2250:
2249:
2241:
2227:
2225:
2224:
2219:
2199:
2198:
2194:
2189:
2188:
2180:
2160:
2159:
2155:
2150:
2149:
2141:
2081:
2079:
2078:
2073:
2058:
2056:
2055:
2050:
2048:
2047:
2024:
2022:
2021:
2016:
2014:
2013:
2005:
1995:
1993:
1992:
1987:
1985:
1984:
1976:
1962:
1960:
1959:
1954:
1934:
1933:
1929:
1921:
1920:
1912:
1906:
1905:
1897:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1798:
1796:
1795:
1790:
1788:
1787:
1782:
1773:
1762:
1761:
1753:
1743:
1741:
1740:
1735:
1733:
1731:
1730:
1729:
1716:
1715:
1706:
1704:
1702:
1694:
1693:
1688:
1682:
1674:
1673:
1665:
1652:
1650:
1649:
1644:
1632:
1630:
1629:
1624:
1612:
1610:
1609:
1604:
1574:
1572:
1571:
1566:
1555:
1554:
1546:
1540:
1539:
1531:
1519:
1518:
1513:
1504:
1493:
1491:
1490:
1489:
1476:
1472:
1471:
1461:
1459:
1457:
1449:
1448:
1443:
1437:
1429:
1427:
1419:
1411:
1389:
1387:
1386:
1381:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1326:
1324:
1323:
1318:
1316:
1312:
1281:
1279:
1278:
1273:
1250:
1249:
1241:
1224:
1222:
1221:
1216:
1196:
1195:
1190:
1189:
1181:
1177:
1173:
1169:
1168:
1159:
1158:
1140:
1135:
1134:
1133:
1120:
1083:
1082:
1074:
1057:
1055:
1054:
1049:
1047:
1046:
1026:
1024:
1023:
1018:
1001:
1000:
955:
954:
944:
939:
909:
908:
903:
902:
894:
880:
878:
877:
872:
870:
869:
852:frequency domain
846:
844:
843:
838:
836:
818:
816:
815:
810:
805:
802:
797:
796:
791:
773:
758:
756:
755:
750:
727:
723:
719:
718:
713:
689:
643:
642:
623:
621:
620:
615:
603:
601:
600:
595:
573:
571:
570:
565:
557:
556:
548:
539:
538:
533:
524:
510:
508:
500:
499:
498:
485:
470:
468:
467:
462:
454:
453:
445:
439:
437:
436:
435:
422:
418:
417:
407:
405:
400:
399:
398:
385:
377:
375:
367:
366:
365:
352:
340:
338:
337:
332:
320:
318:
317:
312:
300:
298:
297:
292:
262:
260:
259:
254:
243:
242:
234:
228:
227:
219:
207:
206:
201:
192:
178:
176:
175:
174:
161:
157:
156:
146:
144:
139:
138:
137:
124:
116:
114:
106:
98:
50:frequency domain
3732:
3731:
3727:
3726:
3725:
3723:
3722:
3721:
3702:
3701:
3700:
3695:
3664:Galerkin method
3607:Method of lines
3584:
3552:Neumann–Neumann
3506:
3463:
3405:
3372:High-resolution
3349:
3320:
3282:
3249:
3244:
3182:
3177:
3176:
3149:J. Comput. Phys
3146:
3145:
3141:
3134:
3121:
3120:
3116:
3074:
3069:
3068:
3064:
3018:
3017:
3013:
2948:Erkintalo, Miro
2946:
2945:
2941:
2936:
2891:
2890:
2869:
2861:
2860:
2789:
2771:
2757:
2741:
2706:
2705:
2683:
2682:
2615:
2597:
2583:
2578:
2577:
2546:
2545:
2517:
2516:
2489:
2475:
2470:
2469:
2394:
2389:
2388:
2365:
2364:
2345:
2335:
2334:
2312:
2311:
2283:
2282:
2263:
2262:
2234:
2233:
2161:
2122:
2087:
2086:
2061:
2060:
2039:
2031:
2030:
1998:
1997:
1969:
1968:
1878:
1852:
1851:
1821:
1820:
1801:
1800:
1777:
1746:
1745:
1721:
1717:
1707:
1683:
1658:
1657:
1635:
1634:
1615:
1614:
1580:
1579:
1508:
1481:
1477:
1463:
1462:
1438:
1420:
1412:
1399:
1398:
1369:
1368:
1349:
1348:
1329:
1328:
1296:
1292:
1284:
1283:
1234:
1233:
1178:
1160:
1150:
1125:
1121:
1118:
1114:
1067:
1066:
1038:
1033:
1032:
992:
946:
891:
886:
885:
861:
856:
855:
821:
820:
786:
764:
763:
708:
678:
674:
634:
629:
628:
606:
605:
604:is taken along
586:
585:
528:
501:
490:
486:
479:
478:
427:
423:
409:
408:
390:
386:
368:
357:
353:
346:
345:
323:
322:
303:
302:
268:
267:
196:
166:
162:
148:
147:
129:
125:
107:
99:
92:
91:
82:
34:pseudo-spectral
12:
11:
5:
3730:
3728:
3720:
3719:
3714:
3704:
3703:
3697:
3696:
3694:
3693:
3688:
3683:
3678:
3673:
3672:
3671:
3661:
3656:
3651:
3646:
3641:
3636:
3635:
3634:
3624:
3619:
3614:
3609:
3604:
3601:Pseudospectral
3598:
3592:
3590:
3586:
3585:
3583:
3582:
3577:
3571:
3565:
3559:
3554:
3549:
3544:
3543:
3542:
3537:
3527:
3522:
3516:
3514:
3508:
3507:
3505:
3504:
3498:
3492:
3486:
3480:
3473:
3471:
3465:
3464:
3462:
3461:
3455:
3450:
3444:
3439:
3433:
3427:
3421:
3415:
3413:
3411:Finite element
3407:
3406:
3404:
3403:
3397:
3391:
3389:Riemann solver
3386:
3380:
3374:
3369:
3363:
3361:
3355:
3354:
3351:
3350:
3348:
3347:
3341:
3335:
3328:
3326:
3322:
3321:
3319:
3318:
3313:
3308:
3303:
3298:
3296:Lax–Friedrichs
3292:
3290:
3284:
3283:
3281:
3280:
3278:Crank–Nicolson
3275:
3268:
3266:
3257:
3251:
3250:
3245:
3243:
3242:
3235:
3228:
3220:
3214:
3213:
3207:
3201:
3195:
3189:
3181:
3178:
3175:
3174:
3155:(2): 203–230.
3139:
3132:
3114:
3085:(2): 231–243.
3062:
3021:Optics Letters
3011:
2952:Optics Letters
2938:
2937:
2935:
2932:
2901:
2898:
2876:
2872:
2868:
2856:
2855:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2814:
2811:
2805:
2802:
2799:
2796:
2792:
2788:
2785:
2778:
2774:
2770:
2767:
2764:
2760:
2756:
2751:
2748:
2744:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2690:
2679:
2678:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2640:
2637:
2631:
2628:
2625:
2622:
2618:
2614:
2611:
2604:
2600:
2596:
2593:
2590:
2586:
2559:
2556:
2530:
2527:
2513:
2512:
2496:
2492:
2488:
2485:
2482:
2478:
2463:
2462:
2450:
2447:
2444:
2441:
2438:
2435:
2430:
2426:
2419:
2416:
2410:
2407:
2404:
2401:
2397:
2372:
2351:
2348:
2343:
2322:
2319:
2296:
2293:
2270:
2247:
2244:
2230:
2229:
2217:
2214:
2211:
2208:
2205:
2202:
2197:
2193:
2186:
2183:
2177:
2174:
2171:
2168:
2164:
2158:
2154:
2147:
2144:
2138:
2135:
2132:
2129:
2125:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2071:
2068:
2046:
2042:
2038:
2011:
2008:
1982:
1979:
1965:
1964:
1952:
1949:
1946:
1943:
1940:
1937:
1932:
1928:
1924:
1918:
1915:
1909:
1903:
1900:
1894:
1891:
1888:
1885:
1881:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1845:
1844:
1828:
1808:
1786:
1781:
1776:
1772:
1768:
1765:
1759:
1756:
1728:
1724:
1720:
1714:
1710:
1701:
1698:
1692:
1687:
1680:
1677:
1671:
1668:
1642:
1622:
1602:
1599:
1596:
1593:
1590:
1587:
1576:
1575:
1564:
1561:
1558:
1552:
1549:
1543:
1537:
1534:
1528:
1525:
1522:
1517:
1512:
1507:
1503:
1499:
1496:
1488:
1484:
1480:
1475:
1470:
1466:
1456:
1453:
1447:
1442:
1435:
1432:
1426:
1423:
1418:
1415:
1409:
1406:
1379:
1376:
1356:
1336:
1315:
1311:
1308:
1305:
1302:
1299:
1295:
1291:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1247:
1244:
1228:By taking the
1226:
1225:
1214:
1211:
1208:
1205:
1202:
1199:
1194:
1187:
1184:
1176:
1172:
1167:
1163:
1157:
1153:
1149:
1146:
1143:
1138:
1132:
1128:
1124:
1117:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1080:
1077:
1045:
1041:
1029:
1028:
1016:
1013:
1010:
1007:
1004:
999:
995:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
953:
949:
943:
938:
935:
931:
927:
924:
921:
918:
915:
912:
907:
900:
897:
868:
864:
835:
831:
828:
808:
800:
795:
790:
785:
782:
779:
776:
772:
760:
759:
748:
745:
742:
739:
736:
733:
730:
726:
722:
717:
712:
707:
704:
701:
698:
695:
692:
688:
684:
681:
677:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
641:
637:
613:
593:
575:
574:
563:
560:
554:
551:
545:
542:
537:
532:
527:
523:
519:
516:
513:
507:
504:
497:
493:
489:
472:
471:
460:
457:
451:
448:
442:
434:
430:
426:
421:
416:
412:
403:
397:
393:
389:
383:
380:
374:
371:
364:
360:
356:
330:
310:
290:
287:
284:
281:
278:
275:
264:
263:
252:
249:
246:
240:
237:
231:
225:
222:
216:
213:
210:
205:
200:
195:
191:
187:
184:
181:
173:
169:
165:
160:
155:
151:
142:
136:
132:
128:
122:
119:
113:
110:
105:
102:
81:
78:
13:
10:
9:
6:
4:
3:
2:
3729:
3718:
3715:
3713:
3710:
3709:
3707:
3692:
3689:
3687:
3684:
3682:
3679:
3677:
3674:
3670:
3667:
3666:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3633:
3630:
3629:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3602:
3599:
3597:
3594:
3593:
3591:
3587:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3541:
3538:
3536:
3533:
3532:
3531:
3528:
3526:
3523:
3521:
3518:
3517:
3515:
3513:
3509:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3474:
3472:
3470:
3466:
3459:
3456:
3454:
3451:
3448:
3445:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3420:
3417:
3416:
3414:
3412:
3408:
3401:
3398:
3395:
3392:
3390:
3387:
3384:
3381:
3378:
3375:
3373:
3370:
3368:
3365:
3364:
3362:
3360:
3359:Finite volume
3356:
3345:
3342:
3339:
3336:
3333:
3330:
3329:
3327:
3323:
3317:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3293:
3291:
3289:
3285:
3279:
3276:
3273:
3270:
3269:
3267:
3265:
3261:
3258:
3256:
3252:
3248:
3241:
3236:
3234:
3229:
3227:
3222:
3221:
3218:
3212:
3208:
3206:
3202:
3200:
3196:
3194:
3190:
3188:
3184:
3183:
3179:
3170:
3166:
3162:
3158:
3154:
3150:
3143:
3140:
3135:
3133:0-12-045143-3
3129:
3125:
3118:
3115:
3110:
3106:
3101:
3096:
3092:
3088:
3084:
3080:
3079:Nanophotonics
3073:
3066:
3063:
3058:
3054:
3050:
3046:
3042:
3038:
3034:
3030:
3026:
3022:
3015:
3012:
3007:
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2966:
2961:
2957:
2953:
2949:
2943:
2940:
2933:
2931:
2929:
2925:
2924:
2919:
2915:
2899:
2896:
2874:
2870:
2866:
2832:
2829:
2826:
2820:
2809:
2803:
2800:
2797:
2794:
2790:
2783:
2776:
2772:
2768:
2765:
2762:
2758:
2749:
2746:
2742:
2738:
2732:
2729:
2726:
2723:
2720:
2717:
2711:
2704:
2703:
2702:
2688:
2658:
2655:
2652:
2646:
2635:
2629:
2626:
2623:
2620:
2616:
2609:
2602:
2598:
2594:
2591:
2588:
2584:
2576:
2575:
2574:
2554:
2525:
2494:
2490:
2486:
2483:
2480:
2476:
2468:
2467:
2466:
2445:
2442:
2439:
2433:
2424:
2414:
2408:
2405:
2402:
2399:
2395:
2387:
2386:
2385:
2370:
2349:
2320:
2317:
2291:
2268:
2242:
2212:
2209:
2206:
2200:
2191:
2181:
2175:
2172:
2169:
2166:
2162:
2152:
2142:
2136:
2133:
2130:
2127:
2123:
2119:
2113:
2110:
2107:
2104:
2101:
2098:
2092:
2085:
2084:
2083:
2069:
2066:
2044:
2040:
2036:
2028:
2006:
1977:
1947:
1944:
1941:
1935:
1926:
1913:
1907:
1898:
1889:
1886:
1883:
1879:
1875:
1869:
1866:
1863:
1857:
1850:
1849:
1848:
1842:
1806:
1784:
1774:
1766:
1763:
1754:
1726:
1722:
1712:
1699:
1696:
1690:
1678:
1675:
1666:
1656:
1655:
1654:
1653:. Note that
1640:
1620:
1597:
1594:
1591:
1585:
1562:
1559:
1547:
1541:
1532:
1523:
1520:
1515:
1505:
1497:
1494:
1486:
1482:
1473:
1468:
1454:
1451:
1445:
1433:
1430:
1424:
1416:
1404:
1397:
1396:
1395:
1391:
1377:
1374:
1354:
1334:
1313:
1309:
1306:
1303:
1300:
1297:
1293:
1289:
1266:
1263:
1260:
1257:
1254:
1242:
1231:
1212:
1206:
1203:
1200:
1192:
1182:
1174:
1170:
1165:
1155:
1151:
1147:
1144:
1136:
1130:
1126:
1122:
1115:
1111:
1108:
1105:
1099:
1096:
1093:
1090:
1087:
1075:
1065:
1064:
1063:
1061:
1043:
1039:
1014:
1011:
1005:
997:
993:
989:
986:
980:
974:
971:
965:
962:
959:
951:
947:
933:
929:
925:
919:
916:
913:
905:
895:
884:
883:
882:
866:
862:
853:
848:
829:
826:
806:
798:
793:
780:
774:
746:
740:
737:
734:
728:
724:
720:
715:
702:
699:
696:
690:
682:
679:
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668:
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126:
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63:
57:
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27:
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3717:Fiber optics
3483:Peridynamics
3301:Lax–Wendroff
3152:
3148:
3142:
3123:
3117:
3082:
3078:
3065:
3024:
3020:
3014:
2958:(1): 37–39.
2955:
2951:
2942:
2921:
2857:
2680:
2514:
2464:
2231:
1966:
1846:
1577:
1392:
1282:one obtains
1227:
1030:
849:
761:
583:
576:
473:
265:
83:
64:dynamics in
58:
29:
25:
21:
15:
3617:Collocation
1799:, and that
54:time domain
3706:Categories
3306:MacCormack
3288:Hyperbolic
2934:References
22:split-step
3622:Level-set
3612:Multigrid
3562:Balancing
3264:Parabolic
3109:2192-8606
3049:1539-4794
2990:1539-4794
2965:1211.1697
2918:algorithm
2821:ψ
2813:^
2795:−
2747:−
2712:ψ
2647:ψ
2639:^
2621:−
2558:^
2529:^
2434:ψ
2429:ℏ
2418:^
2400:−
2347:∂
2342:∂
2295:^
2246:^
2201:ψ
2196:ℏ
2185:^
2167:−
2157:ℏ
2146:^
2128:−
2120:≈
2093:ψ
2010:^
1981:^
1936:ψ
1931:ℏ
1917:^
1902:^
1884:−
1858:ψ
1827:ℏ
1775:ψ
1767:γ
1758:^
1719:∂
1709:∂
1686:ℏ
1679:−
1670:^
1633:and time
1586:ψ
1560:ψ
1551:^
1536:^
1521:ψ
1506:ψ
1498:γ
1479:∂
1474:ψ
1465:∂
1441:ℏ
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1422:∂
1417:ψ
1414:∂
1408:ℏ
1255:ω
1246:~
1201:ω
1186:~
1152:ω
1148:−
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1127:β
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1088:ω
1079:~
1040:ω
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987:ω
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937:∞
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930:∫
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830:∈
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683:γ
672:
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450:^
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186:γ
164:∂
150:∂
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109:∂
101:∂
40:like the
3596:Spectral
3535:additive
3458:Smoothed
3424:Extended
3057:21808332
2998:23282830
2916:of this
2363:, where
3580:FETI-DP
3460:(S-FEM)
3379:(MUSCL)
3367:Godunov
3157:Bibcode
3087:Bibcode
3029:Bibcode
3006:7248349
2970:Bibcode
1839:is the
74:soliton
26:Fourier
3589:Others
3576:(FETI)
3570:(BDDC)
3442:Mortar
3426:(XFEM)
3419:hp-FEM
3402:(WENO)
3385:(AUSM)
3346:(FDTD)
3340:(FDFD)
3325:Others
3311:Upwind
3274:(FTCS)
3130:
3107:
3055:
3047:
3004:
2996:
2988:
2912:. The
2681:where
1967:Since
1578:where
1031:where
881:using
266:where
30:method
20:, the
3603:(DVR)
3564:(BDD)
3503:(PIC)
3497:(MPM)
3491:(MPS)
3479:(SPH)
3449:(GDM)
3438:(SEM)
3396:(ENO)
3334:(ADI)
3075:(PDF)
3002:S2CID
2960:arXiv
803:const
32:is a
3485:(PD)
3432:(DG)
3128:ISBN
3105:ISSN
3053:PMID
3045:ISSN
2994:PMID
2986:ISSN
2544:and
2333:for
1996:and
1744:and
3165:doi
3095:doi
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2978:doi
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1109:exp
972:exp
669:exp
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