2221:
1554:
2216:{\displaystyle {\begin{aligned}x_{i}^{1}&=x_{i}+c_{1}\,v_{i}\,\Delta t,\\v_{i}^{1}&=v_{i}+d_{1}\,a(x_{i}^{1})\,\Delta t,\\x_{i}^{2}&=x_{i}^{1}+c_{2}\,v_{i}^{1}\,\Delta t,\\v_{i}^{2}&=v_{i}^{1}+d_{2}\,a(x_{i}^{2})\,\Delta t,\\x_{i}^{3}&=x_{i}^{2}+c_{3}\,v_{i}^{2}\,\Delta t,\\v_{i}^{3}&=v_{i}^{2}+d_{3}\,a(x_{i}^{3})\,\Delta t,\\x_{i+1}&\equiv x_{i}^{4}=x_{i}^{3}+c_{4}\,v_{i}^{3}\,\Delta t,\\v_{i+1}&\equiv v_{i}^{4}=v_{i}^{3}\\\end{aligned}}}
2914:
2601:
1432:
1142:
257:
661:
2909:{\displaystyle {\begin{aligned}w_{0}&\equiv -{\frac {\sqrt{2}}{2-{\sqrt{2}}}},\\w_{1}&\equiv {\frac {1}{2-{\sqrt{2}}}},\\c_{1}&=c_{4}\equiv {\frac {w_{1}}{2}},c_{2}=c_{3}\equiv {\frac {w_{0}+w_{1}}{2}},\\d_{1}&=d_{3}\equiv w_{1},d_{2}\equiv w_{0}\\\end{aligned}}}
1179:
928:
1536:. In this approach, the leapfrog is applied over a number of different timesteps. It turns out that when the correct timesteps are used in sequence, the errors cancel and far higher order integrators can be easily produced.
243:
1544:
One step under the 4th order
Yoshida integrator requires four intermediary steps. The position and velocity are computed at different times. Only three (computationally expensive) acceleration calculations are required.
446:
1481:
One use of this equation is in gravity simulations, since in that case the acceleration depends only on the positions of the gravitating masses (and not on their velocities), although higher-order integrators (such as
1437:
which is primarily used where variable time-steps are required. The separation of the acceleration calculation onto the beginning and end of a step means that if time resolution is increased by a factor of two
128:
2606:
1559:
1184:
933:
451:
1476:
1427:{\displaystyle {\begin{aligned}v_{i+1/2}&=v_{i}+a_{i}{\frac {\Delta t}{2}},\\x_{i+1}&=x_{i}+v_{i+1/2}\Delta t,\\v_{i+1}&=v_{i+1/2}+a_{i+1}{\frac {\Delta t}{2}},\end{aligned}}}
2534:
1137:{\displaystyle {\begin{aligned}x_{i+1}&=x_{i}+v_{i}\,\Delta t+{\tfrac {1}{2}}\,a_{i}\,\Delta t^{\,2},\\v_{i+1}&=v_{i}+{\tfrac {1}{2}}(a_{i}+a_{i+1})\,\Delta t.\end{aligned}}}
427:
369:, which is only first-order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step
3531:
3478:
3445:
3048:
2994:
2593:
2314:
352:
3050:. Please note that position and velocity are computed at different times and some intermediary steps are backwards in time. To illustrate this, we give the numerical values of
2459:
857:
3213:
3180:
3144:
3108:
2375:
2264:
753:
2407:
2940:
1168:
920:
390:
133:
808:
3075:
691:
299:
2334:
1509:) energy of a Hamiltonian dynamical system. This is especially useful when computing orbital dynamics, as many other integration schemes, such as the (order-4)
897:
877:
773:
711:
260:
Comparison of Euler's and
Leapfrog integration energy conserving properties for N bodies orbiting a point source mass. Same time-step used in both simulations.
3437:
3223:
39:
432:
Using
Yoshida coefficients, applying the leapfrog integrator multiple times with the correct timesteps, a much higher order integrator can be generated.
656:{\displaystyle {\begin{aligned}a_{i}&=A(x_{i}),\\v_{i+1/2}&=v_{i-1/2}+a_{i}\,\Delta t,\\x_{i+1}&=x_{i}+v_{i+1/2}\,\Delta t,\end{aligned}}}
3428:
1533:
49:
3611:
3496:
3338:
3421:
922:
is the size of each time step. These equations can be expressed in a form that gives velocity at integer steps as well:
3309:
1441:
3564:
3401:
3549:
17:
3414:
2467:
1524:, a method for drawing random samples from a probability distribution whose overall normalization is unknown.
3354:
1489:
There are two primary strengths to leapfrog integration when applied to mechanics problems. The first is the
395:
362:
2539:
2269:
3406:
2999:
2945:
304:
3463:
264:
The method is known by different names in different disciplines. In particular, it is similar to the
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3287:
2412:
813:
3468:
3185:
3149:
3113:
3080:
2339:
3554:
3544:
3243:
2229:
1510:
1483:
716:
3559:
3539:
2380:
1532:
The leapfrog integrator can be converted into higher order integrators using techniques due to
2922:
1150:
902:
372:
3590:
3501:
3458:
3228:
1517:
1506:
1502:
778:
43:
3053:
669:
275:
8:
3516:
1478:), then only one extra (computationally expensive) acceleration calculation is required.
250:
3486:
3238:
2319:
1521:
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882:
862:
758:
696:
269:
31:
1513:
method, do not conserve energy and allow the system to drift substantially over time.
3384:
3334:
3305:
3275:
3233:
366:
3511:
3380:
3286:
Skeel, R. D., "Variable Step Size
Destabilizes the Stömer/Leapfrog/Verlet Method",
246:
3506:
3491:
3330:
238:{\displaystyle {\dot {v}}={\frac {dv}{dt}}=A(x),\;{\dot {x}}={\frac {dx}{dt}}=v,}
1548:
The equations for the 4th order integrator to update position and velocity are
3371:
Yoshida, Haruo (1990). "Construction of higher order symplectic integrators".
440:
In leapfrog integration, the equations for updating position and velocity are
3605:
1497:
steps, and then reverse the direction of integration and integrate backwards
256:
3453:
1501:
steps to arrive at the same starting position. The second strength is its
3436:
354:
at different interleaved time points, staggered in such a way that they "
1173:
The synchronised form can be re-arranged to the 'kick-drift-kick' form;
2461:
are the final position and velocity under one 4th order
Yoshida step.
1505:
nature, which implies that it conserves the (slightly modified; see
355:
2316:
are intermediary position and velocity at intermediary step
272:. Leapfrog integration is equivalent to updating positions
123:{\displaystyle {\ddot {x}}={\frac {d^{2}x}{dt^{2}}}=A(x),}
3304:(1 ed.). Oxford University Press. pp. 121–124.
1516:
Because of its time-reversibility, and because it is a
3302:
Statistical
Mechanics: Theory and Molecular Simulation
3002:
2948:
1074:
993:
3438:
Numerical methods for ordinary differential equations
3224:
Numerical methods for ordinary differential equations
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136:
52:
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step which implies that coefficients sum up to one:
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2908:
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2528:
2453:
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2328:
2308:
2258:
2215:
1493:of the Leapfrog method. One can integrate forward
1470:
1426:
1162:
1147:However, in this synchronized form, the time-step
1136:
914:
891:
871:
851:
802:
767:
747:
705:
685:
655:
421:
384:
346:
293:
237:
122:
3603:
1539:
1471:{\displaystyle \Delta t\rightarrow \Delta t/2}
3422:
3366:
3364:
859:is the acceleration, or second derivative of
3429:
3415:
3361:
190:
3299:
2595:are derived in (see the equation (4.6))
2529:{\displaystyle (c_{1},c_{2},c_{3},c_{4})}
2139:
2123:
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2015:
1951:
1935:
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1846:
1782:
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1702:
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1618:
1607:
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1024:
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1004:
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639:
565:
3327:Pattern Recognition and Machine Learning
3266:, McGraw-Hill Book Company, 1985, p. 56.
2266:are the starting position and velocity,
1170:must be constant to maintain stability.
755:is the velocity, or first derivative of
255:
3569:
3370:
3264:Plasma Physics via Computer Simulations
1520:, leapfrog integration is also used in
14:
3604:
3324:
3410:
1527:
422:{\displaystyle \Delta t<2/\omega }
2377:is the acceleration at the position
3043:{\textstyle \sum _{i=1}^{3}d_{i}=1}
2989:{\textstyle \sum _{i=1}^{4}c_{i}=1}
2588:{\displaystyle (d_{1},d_{2},d_{3})}
2309:{\displaystyle x_{i}^{n},v_{i}^{n}}
24:
3276:4.1 Two Ways to Write the Leapfrog
3262:C. K. Birdsall and A. B. Langdon,
2926:
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2041:
1952:
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1783:
1703:
1619:
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1445:
1405:
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1016:
983:
906:
640:
566:
399:
376:
347:{\displaystyle v(t)={\dot {x}}(t)}
25:
3623:
3395:
3612:Numerical differential equations
3565:Backward differentiation formula
2919:All intermediary steps form one
2454:{\displaystyle x_{i+1},v_{i+1}}
3347:
3318:
3293:
3280:
3269:
3256:
2582:
2543:
2523:
2471:
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2346:
2037:
2019:
1868:
1850:
1699:
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1085:
852:{\displaystyle a_{i}=A(x_{i})}
846:
833:
487:
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341:
335:
317:
311:
288:
282:
268:method, which is a variant of
245:particularly in the case of a
184:
178:
114:
108:
13:
1:
3249:
3208:{\displaystyle c_{4}=0.6756.}
3175:{\displaystyle c_{3}=-0.1756}
3139:{\displaystyle c_{2}=-0.1756}
3385:10.1016/0375-9601(90)90092-3
3325:Bishop, Christopher (2006).
3290:, Vol. 33, 1993, p. 172–175.
3103:{\displaystyle c_{1}=0.6756}
2370:{\displaystyle a(x_{i}^{n})}
1540:4th order Yoshida integrator
1486:) are more frequently used.
435:
130:or equivalently of the form
42:for numerically integrating
7:
3550:List of Runge–Kutta methods
3403:, Drexel University Physics
3300:Tuckerman, Mark E. (2010).
3217:
2259:{\displaystyle x_{i},v_{i}}
748:{\displaystyle v_{i+1/2\,}}
10:
3628:
361:Leapfrog integration is a
3583:
3530:
3477:
3444:
3288:BIT Numerical Mathematics
2402:{\displaystyle x_{i}^{n}}
2935:{\displaystyle \Delta t}
1163:{\displaystyle \Delta t}
915:{\displaystyle \Delta t}
385:{\displaystyle \Delta t}
3555:Linear multistep method
3244:Runge–Kutta integration
1522:Hamiltonian Monte Carlo
803:{\displaystyle i+1/2\,}
365:method, in contrast to
3560:General linear methods
3540:Exponential integrator
3229:Symplectic integration
3209:
3176:
3140:
3104:
3071:
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3023:
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423:
386:
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261:
239:
124:
44:differential equations
3591:Symplectic integrator
3575:Gauss–Legendre method
3210:
3177:
3141:
3105:
3072:
3070:{\displaystyle c_{n}}
3045:
3003:
2991:
2949:
2937:
2911:
2590:
2531:
2456:
2404:
2372:
2331:
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2218:
1518:symplectic integrator
1507:symplectic integrator
1473:
1429:
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1139:
917:
894:
874:
854:
805:
770:
750:
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688:
686:{\displaystyle x_{i}}
658:
424:
387:
349:
296:
259:
240:
125:
3532:Higher-order methods
3522:Leapfrog integration
3479:Second-order methods
3333:. pp. 548–554.
3186:
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3114:
3081:
3054:
3000:
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2602:
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2413:
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1151:
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883:
863:
814:
779:
759:
717:
697:
693:is position at step
670:
447:
396:
373:
305:
294:{\displaystyle x(t)}
276:
134:
50:
36:leapfrog integration
3545:Runge–Kutta methods
3517:Newmark-beta method
3464:Semi-implicit Euler
3446:First-order methods
2398:
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2305:
2287:
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2190:
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2001:
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1752:
1730:
1698:
1646:
1576:
1484:Runge–Kutta methods
358:" over each other.
251:classical mechanics
27:Mathematics concept
3502:Beeman's algorithm
3487:Verlet integration
3239:Verlet integration
3205:
3172:
3136:
3100:
3067:
3040:
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1562:
1528:Yoshida algorithms
1491:time-reversibility
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270:Verlet integration
262:
235:
120:
32:numerical analysis
3599:
3598:
3469:Exponential Euler
3373:Physics Letters A
3340:978-0-387-31073-2
3234:Euler integration
2827:
2766:
2712:
2709:
2661:
2658:
2640:
2329:{\displaystyle n}
1415:
1254:
1082:
1001:
892:{\displaystyle i}
872:{\displaystyle x}
768:{\displaystyle x}
706:{\displaystyle i}
392:is constant, and
367:Euler integration
332:
224:
200:
170:
146:
100:
62:
16:(Redirected from
3619:
3497:Trapezoidal rule
3431:
3424:
3417:
3408:
3407:
3389:
3388:
3379:(5–7): 262–268.
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3507:Midpoint method
3492:Velocity Verlet
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3331:Springer-Verlag
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2721:
2718:
2717:
2704:
2699:
2692:
2687:
2680:
2674:
2670:
2667:
2666:
2653:
2648:
2641:
2635:
2629:
2619:
2613:
2609:
2605:
2603:
2600:
2599:
2576:
2572:
2563:
2559:
2550:
2546:
2541:
2538:
2537:
2517:
2513:
2504:
2500:
2491:
2487:
2478:
2474:
2469:
2466:
2465:
2439:
2435:
2420:
2416:
2414:
2411:
2410:
2393:
2388:
2382:
2379:
2378:
2358:
2353:
2341:
2338:
2337:
2321:
2318:
2317:
2300:
2295:
2282:
2277:
2271:
2268:
2267:
2250:
2246:
2237:
2233:
2231:
2228:
2227:
2210:
2209:
2203:
2198:
2185:
2180:
2169:
2157:
2153:
2150:
2149:
2133:
2128:
2117:
2113:
2104:
2099:
2086:
2081:
2070:
2058:
2054:
2051:
2050:
2031:
2026:
2009:
2005:
1996:
1991:
1980:
1974:
1969:
1962:
1961:
1945:
1940:
1929:
1925:
1916:
1911:
1900:
1894:
1889:
1882:
1881:
1862:
1857:
1840:
1836:
1827:
1822:
1811:
1805:
1800:
1793:
1792:
1776:
1771:
1760:
1756:
1747:
1742:
1731:
1725:
1720:
1713:
1712:
1693:
1688:
1671:
1667:
1658:
1654:
1647:
1641:
1636:
1629:
1628:
1612:
1608:
1601:
1597:
1588:
1584:
1577:
1571:
1566:
1558:
1556:
1553:
1552:
1542:
1530:
1460:
1443:
1440:
1439:
1421:
1420:
1404:
1402:
1390:
1386:
1373:
1363:
1359:
1352:
1340:
1336:
1333:
1332:
1313:
1303:
1299:
1290:
1286:
1279:
1267:
1263:
1260:
1259:
1243:
1241:
1235:
1231:
1222:
1218:
1211:
1201:
1191:
1187:
1183:
1181:
1178:
1177:
1152:
1149:
1148:
1131:
1130:
1105:
1101:
1092:
1088:
1073:
1064:
1060:
1053:
1041:
1037:
1034:
1033:
1023:
1019:
1009:
1005:
992:
976:
972:
963:
959:
952:
940:
936:
932:
930:
927:
926:
904:
901:
900:
884:
881:
880:
864:
861:
860:
840:
836:
821:
817:
815:
812:
811:
791:
780:
777:
776:
760:
757:
756:
734:
724:
720:
718:
715:
714:
698:
695:
694:
677:
673:
671:
668:
667:
650:
649:
629:
619:
615:
606:
602:
595:
583:
579:
576:
575:
559:
555:
542:
532:
528:
521:
511:
501:
497:
494:
493:
481:
477:
464:
458:
454:
450:
448:
445:
444:
438:
411:
397:
394:
393:
374:
371:
370:
324:
323:
306:
303:
302:
301:and velocities
277:
274:
273:
266:velocity Verlet
216:
208:
206:
192:
191:
162:
154:
152:
138:
137:
135:
132:
131:
93:
89:
85:
75:
71:
70:
68:
54:
53:
51:
48:
47:
28:
23:
22:
18:Leapfrog method
15:
12:
11:
5:
3625:
3615:
3614:
3597:
3596:
3594:
3593:
3587:
3585:
3581:
3580:
3578:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3536:
3534:
3528:
3527:
3525:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3483:
3481:
3475:
3474:
3472:
3471:
3466:
3461:
3459:Backward Euler
3456:
3450:
3448:
3442:
3441:
3434:
3433:
3426:
3419:
3411:
3405:
3404:
3397:
3396:External links
3394:
3391:
3390:
3360:
3346:
3339:
3317:
3310:
3292:
3279:
3268:
3254:
3253:
3251:
3248:
3247:
3246:
3241:
3236:
3231:
3226:
3219:
3216:
3204:
3201:
3196:
3192:
3171:
3168:
3165:
3160:
3156:
3135:
3132:
3129:
3124:
3120:
3099:
3096:
3091:
3087:
3077:coefficients:
3064:
3060:
3039:
3036:
3031:
3027:
3021:
3016:
3013:
3010:
3006:
2985:
2982:
2977:
2973:
2967:
2962:
2959:
2956:
2952:
2931:
2928:
2917:
2916:
2899:
2895:
2891:
2886:
2882:
2878:
2873:
2869:
2865:
2860:
2856:
2852:
2849:
2847:
2843:
2839:
2835:
2834:
2831:
2826:
2820:
2816:
2812:
2807:
2803:
2796:
2791:
2787:
2783:
2778:
2774:
2770:
2765:
2760:
2756:
2750:
2745:
2741:
2737:
2734:
2732:
2728:
2724:
2720:
2719:
2716:
2707:
2703:
2698:
2695:
2691:
2686:
2683:
2681:
2677:
2673:
2669:
2668:
2665:
2656:
2652:
2647:
2644:
2638:
2634:
2628:
2625:
2622:
2620:
2616:
2612:
2608:
2607:
2584:
2579:
2575:
2571:
2566:
2562:
2558:
2553:
2549:
2545:
2525:
2520:
2516:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2481:
2477:
2473:
2448:
2445:
2442:
2438:
2434:
2429:
2426:
2423:
2419:
2396:
2391:
2387:
2366:
2361:
2356:
2352:
2348:
2345:
2325:
2303:
2298:
2294:
2290:
2285:
2280:
2276:
2253:
2249:
2245:
2240:
2236:
2224:
2223:
2206:
2201:
2197:
2193:
2188:
2183:
2179:
2175:
2172:
2170:
2166:
2163:
2160:
2156:
2152:
2151:
2148:
2145:
2142:
2136:
2131:
2127:
2120:
2116:
2112:
2107:
2102:
2098:
2094:
2089:
2084:
2080:
2076:
2073:
2071:
2067:
2064:
2061:
2057:
2053:
2052:
2049:
2046:
2043:
2039:
2034:
2029:
2025:
2021:
2018:
2012:
2008:
2004:
1999:
1994:
1990:
1986:
1983:
1981:
1977:
1972:
1968:
1964:
1963:
1960:
1957:
1954:
1948:
1943:
1939:
1932:
1928:
1924:
1919:
1914:
1910:
1906:
1903:
1901:
1897:
1892:
1888:
1884:
1883:
1880:
1877:
1874:
1870:
1865:
1860:
1856:
1852:
1849:
1843:
1839:
1835:
1830:
1825:
1821:
1817:
1814:
1812:
1808:
1803:
1799:
1795:
1794:
1791:
1788:
1785:
1779:
1774:
1770:
1763:
1759:
1755:
1750:
1745:
1741:
1737:
1734:
1732:
1728:
1723:
1719:
1715:
1714:
1711:
1708:
1705:
1701:
1696:
1691:
1687:
1683:
1680:
1674:
1670:
1666:
1661:
1657:
1653:
1650:
1648:
1644:
1639:
1635:
1631:
1630:
1627:
1624:
1621:
1615:
1611:
1604:
1600:
1596:
1591:
1587:
1583:
1580:
1578:
1574:
1569:
1565:
1561:
1560:
1541:
1538:
1529:
1526:
1467:
1463:
1459:
1456:
1453:
1450:
1447:
1435:
1434:
1419:
1414:
1410:
1407:
1399:
1396:
1393:
1389:
1385:
1380:
1376:
1372:
1369:
1366:
1362:
1358:
1355:
1353:
1349:
1346:
1343:
1339:
1335:
1334:
1331:
1328:
1325:
1320:
1316:
1312:
1309:
1306:
1302:
1298:
1293:
1289:
1285:
1282:
1280:
1276:
1273:
1270:
1266:
1262:
1261:
1258:
1253:
1249:
1246:
1238:
1234:
1230:
1225:
1221:
1217:
1214:
1212:
1208:
1204:
1200:
1197:
1194:
1190:
1186:
1185:
1159:
1156:
1145:
1144:
1129:
1126:
1123:
1119:
1114:
1111:
1108:
1104:
1100:
1095:
1091:
1087:
1081:
1078:
1072:
1067:
1063:
1059:
1056:
1054:
1050:
1047:
1044:
1040:
1036:
1035:
1032:
1027:
1022:
1018:
1012:
1008:
1000:
997:
991:
988:
985:
979:
975:
971:
966:
962:
958:
955:
953:
949:
946:
943:
939:
935:
934:
911:
908:
888:
868:
848:
843:
839:
835:
832:
829:
824:
820:
798:
794:
790:
787:
784:
764:
741:
737:
733:
730:
727:
723:
702:
680:
676:
664:
663:
648:
645:
642:
636:
632:
628:
625:
622:
618:
614:
609:
605:
601:
598:
596:
592:
589:
586:
582:
578:
577:
574:
571:
568:
562:
558:
554:
549:
545:
541:
538:
535:
531:
527:
524:
522:
518:
514:
510:
507:
504:
500:
496:
495:
492:
489:
484:
480:
476:
473:
470:
467:
465:
461:
457:
453:
452:
437:
434:
418:
414:
410:
407:
404:
401:
381:
378:
343:
340:
337:
331:
328:
322:
319:
316:
313:
310:
290:
287:
284:
281:
234:
231:
228:
222:
219:
214:
211:
205:
199:
196:
189:
186:
183:
180:
177:
174:
168:
165:
160:
157:
151:
145:
142:
119:
116:
113:
110:
107:
104:
96:
92:
88:
83:
78:
74:
67:
61:
58:
26:
9:
6:
4:
3:
2:
3624:
3613:
3610:
3609:
3607:
3592:
3589:
3588:
3586:
3582:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3537:
3535:
3533:
3529:
3523:
3520:
3518:
3515:
3513:
3512:Heun's method
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3482:
3480:
3476:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3451:
3449:
3447:
3443:
3439:
3432:
3427:
3425:
3420:
3418:
3413:
3412:
3409:
3402:
3400:
3399:
3386:
3382:
3378:
3374:
3367:
3365:
3356:
3355:"./Ch07.HTML"
3350:
3342:
3336:
3332:
3328:
3321:
3313:
3311:9780198525264
3307:
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3296:
3289:
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3277:
3272:
3265:
3259:
3255:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3221:
3215:
3202:
3199:
3194:
3190:
3169:
3166:
3163:
3158:
3154:
3133:
3130:
3127:
3122:
3118:
3097:
3094:
3089:
3085:
3062:
3058:
3037:
3034:
3029:
3025:
3019:
3014:
3011:
3008:
3004:
2983:
2980:
2975:
2971:
2965:
2960:
2957:
2954:
2950:
2929:
2897:
2893:
2889:
2884:
2880:
2876:
2871:
2867:
2863:
2858:
2854:
2850:
2848:
2841:
2837:
2829:
2824:
2818:
2814:
2810:
2805:
2801:
2794:
2789:
2785:
2781:
2776:
2772:
2768:
2763:
2758:
2754:
2748:
2743:
2739:
2735:
2733:
2726:
2722:
2714:
2705:
2701:
2696:
2693:
2689:
2684:
2682:
2675:
2671:
2663:
2654:
2650:
2645:
2642:
2636:
2632:
2626:
2623:
2621:
2614:
2610:
2598:
2597:
2596:
2577:
2573:
2569:
2564:
2560:
2556:
2551:
2547:
2518:
2514:
2510:
2505:
2501:
2497:
2492:
2488:
2484:
2479:
2475:
2464:Coefficients
2462:
2446:
2443:
2440:
2436:
2432:
2427:
2424:
2421:
2417:
2394:
2389:
2385:
2359:
2354:
2350:
2343:
2323:
2301:
2296:
2292:
2288:
2283:
2278:
2274:
2251:
2247:
2243:
2238:
2234:
2204:
2199:
2195:
2191:
2186:
2181:
2177:
2173:
2171:
2164:
2161:
2158:
2154:
2146:
2143:
2134:
2129:
2125:
2118:
2114:
2110:
2105:
2100:
2096:
2092:
2087:
2082:
2078:
2074:
2072:
2065:
2062:
2059:
2055:
2047:
2044:
2032:
2027:
2023:
2016:
2010:
2006:
2002:
1997:
1992:
1988:
1984:
1982:
1975:
1970:
1966:
1958:
1955:
1946:
1941:
1937:
1930:
1926:
1922:
1917:
1912:
1908:
1904:
1902:
1895:
1890:
1886:
1878:
1875:
1863:
1858:
1854:
1847:
1841:
1837:
1833:
1828:
1823:
1819:
1815:
1813:
1806:
1801:
1797:
1789:
1786:
1777:
1772:
1768:
1761:
1757:
1753:
1748:
1743:
1739:
1735:
1733:
1726:
1721:
1717:
1709:
1706:
1694:
1689:
1685:
1678:
1672:
1668:
1664:
1659:
1655:
1651:
1649:
1642:
1637:
1633:
1625:
1622:
1613:
1609:
1602:
1598:
1594:
1589:
1585:
1581:
1579:
1572:
1567:
1563:
1551:
1550:
1549:
1546:
1537:
1535:
1534:Haruo Yoshida
1525:
1523:
1519:
1514:
1512:
1508:
1504:
1500:
1496:
1492:
1487:
1485:
1479:
1465:
1461:
1457:
1448:
1417:
1412:
1408:
1397:
1394:
1391:
1387:
1383:
1378:
1374:
1370:
1367:
1364:
1360:
1356:
1354:
1347:
1344:
1341:
1337:
1329:
1326:
1318:
1314:
1310:
1307:
1304:
1300:
1296:
1291:
1287:
1283:
1281:
1274:
1271:
1268:
1264:
1256:
1251:
1247:
1236:
1232:
1228:
1223:
1219:
1215:
1213:
1206:
1202:
1198:
1195:
1192:
1188:
1176:
1175:
1174:
1171:
1157:
1127:
1124:
1112:
1109:
1106:
1102:
1098:
1093:
1089:
1079:
1076:
1070:
1065:
1061:
1057:
1055:
1048:
1045:
1042:
1038:
1030:
1025:
1020:
1010:
1006:
998:
995:
989:
986:
977:
973:
969:
964:
960:
956:
954:
947:
944:
941:
937:
925:
924:
923:
909:
886:
866:
841:
837:
830:
827:
822:
818:
796:
792:
788:
785:
782:
762:
739:
735:
731:
728:
725:
721:
700:
678:
674:
646:
643:
634:
630:
626:
623:
620:
616:
612:
607:
603:
599:
597:
590:
587:
584:
580:
572:
569:
560:
556:
552:
547:
543:
539:
536:
533:
529:
525:
523:
516:
512:
508:
505:
502:
498:
490:
482:
478:
471:
468:
466:
459:
455:
443:
442:
441:
433:
430:
416:
412:
408:
405:
402:
379:
368:
364:
359:
357:
338:
329:
326:
320:
314:
308:
285:
279:
271:
267:
258:
254:
252:
248:
232:
229:
226:
220:
217:
212:
209:
203:
197:
194:
187:
181:
175:
172:
166:
163:
158:
155:
149:
143:
140:
117:
111:
105:
102:
94:
90:
86:
81:
76:
72:
65:
59:
56:
45:
41:
37:
33:
19:
3521:
3454:Euler method
3376:
3372:
3349:
3329:. New York:
3326:
3320:
3301:
3295:
3282:
3271:
3263:
3258:
2918:
2463:
2225:
1547:
1543:
1531:
1515:
1498:
1494:
1488:
1480:
1436:
1172:
1146:
665:
439:
431:
363:second-order
360:
265:
263:
46:of the form
35:
29:
1511:Runge–Kutta
3250:References
1503:symplectic
879:, at step
775:, at step
3167:−
3131:−
3005:∑
2951:∑
2927:Δ
2890:≡
2864:≡
2795:≡
2749:≡
2697:−
2685:≡
2646:−
2627:−
2624:≡
2174:≡
2141:Δ
2075:≡
2042:Δ
1953:Δ
1873:Δ
1784:Δ
1704:Δ
1620:Δ
1455:Δ
1452:→
1446:Δ
1406:Δ
1324:Δ
1245:Δ
1155:Δ
1122:Δ
1017:Δ
984:Δ
907:Δ
641:Δ
567:Δ
537:−
436:Algorithm
417:ω
400:Δ
377:Δ
330:˙
198:˙
144:˙
60:¨
3606:Category
3218:See also
356:leapfrog
3570:Yoshida
3203:0.6756.
3584:Theory
3337:
3308:
3170:0.1756
3134:0.1756
3098:0.6756
2409:, and
2226:where
899:, and
666:where
40:method
38:is a
3335:ISBN
3306:ISBN
2996:and
2536:and
406:<
3381:doi
3377:150
253:.
249:of
30:In
3608::
3375:.
3363:^
3182:,
3146:,
3110:,
2336:,
810:,
713:,
429:.
34:,
3430:e
3423:t
3416:v
3387:.
3383::
3357:.
3343:.
3314:.
3200:=
3195:4
3191:c
3164:=
3159:3
3155:c
3128:=
3123:2
3119:c
3095:=
3090:1
3086:c
3063:n
3059:c
3038:1
3035:=
3030:i
3026:d
3020:3
3015:1
3012:=
3009:i
2984:1
2981:=
2976:i
2972:c
2966:4
2961:1
2958:=
2955:i
2930:t
2898:0
2894:w
2885:2
2881:d
2877:,
2872:1
2868:w
2859:3
2855:d
2851:=
2842:1
2838:d
2830:,
2825:2
2819:1
2815:w
2811:+
2806:0
2802:w
2790:3
2786:c
2782:=
2777:2
2773:c
2769:,
2764:2
2759:1
2755:w
2744:4
2740:c
2736:=
2727:1
2723:c
2715:,
2706:3
2702:2
2694:2
2690:1
2676:1
2672:w
2664:,
2655:3
2651:2
2643:2
2637:3
2633:2
2615:0
2611:w
2583:)
2578:3
2574:d
2570:,
2565:2
2561:d
2557:,
2552:1
2548:d
2544:(
2524:)
2519:4
2515:c
2511:,
2506:3
2502:c
2498:,
2493:2
2489:c
2485:,
2480:1
2476:c
2472:(
2447:1
2444:+
2441:i
2437:v
2433:,
2428:1
2425:+
2422:i
2418:x
2395:n
2390:i
2386:x
2365:)
2360:n
2355:i
2351:x
2347:(
2344:a
2324:n
2302:n
2297:i
2293:v
2289:,
2284:n
2279:i
2275:x
2252:i
2248:v
2244:,
2239:i
2235:x
2205:3
2200:i
2196:v
2192:=
2187:4
2182:i
2178:v
2165:1
2162:+
2159:i
2155:v
2147:,
2144:t
2135:3
2130:i
2126:v
2119:4
2115:c
2111:+
2106:3
2101:i
2097:x
2093:=
2088:4
2083:i
2079:x
2066:1
2063:+
2060:i
2056:x
2048:,
2045:t
2038:)
2033:3
2028:i
2024:x
2020:(
2017:a
2011:3
2007:d
2003:+
1998:2
1993:i
1989:v
1985:=
1976:3
1971:i
1967:v
1959:,
1956:t
1947:2
1942:i
1938:v
1931:3
1927:c
1923:+
1918:2
1913:i
1909:x
1905:=
1896:3
1891:i
1887:x
1879:,
1876:t
1869:)
1864:2
1859:i
1855:x
1851:(
1848:a
1842:2
1838:d
1834:+
1829:1
1824:i
1820:v
1816:=
1807:2
1802:i
1798:v
1790:,
1787:t
1778:1
1773:i
1769:v
1762:2
1758:c
1754:+
1749:1
1744:i
1740:x
1736:=
1727:2
1722:i
1718:x
1710:,
1707:t
1700:)
1695:1
1690:i
1686:x
1682:(
1679:a
1673:1
1669:d
1665:+
1660:i
1656:v
1652:=
1643:1
1638:i
1634:v
1626:,
1623:t
1614:i
1610:v
1603:1
1599:c
1595:+
1590:i
1586:x
1582:=
1573:1
1568:i
1564:x
1499:n
1495:n
1466:2
1462:/
1458:t
1449:t
1438:(
1418:,
1413:2
1409:t
1398:1
1395:+
1392:i
1388:a
1384:+
1379:2
1375:/
1371:1
1368:+
1365:i
1361:v
1357:=
1348:1
1345:+
1342:i
1338:v
1330:,
1327:t
1319:2
1315:/
1311:1
1308:+
1305:i
1301:v
1297:+
1292:i
1288:x
1284:=
1275:1
1272:+
1269:i
1265:x
1257:,
1252:2
1248:t
1237:i
1233:a
1229:+
1224:i
1220:v
1216:=
1207:2
1203:/
1199:1
1196:+
1193:i
1189:v
1158:t
1128:.
1125:t
1118:)
1113:1
1110:+
1107:i
1103:a
1099:+
1094:i
1090:a
1086:(
1080:2
1077:1
1071:+
1066:i
1062:v
1058:=
1049:1
1046:+
1043:i
1039:v
1031:,
1026:2
1021:t
1011:i
1007:a
999:2
996:1
990:+
987:t
978:i
974:v
970:+
965:i
961:x
957:=
948:1
945:+
942:i
938:x
910:t
887:i
867:x
847:)
842:i
838:x
834:(
831:A
828:=
823:i
819:a
797:2
793:/
789:1
786:+
783:i
763:x
740:2
736:/
732:1
729:+
726:i
722:v
701:i
679:i
675:x
647:,
644:t
635:2
631:/
627:1
624:+
621:i
617:v
613:+
608:i
604:x
600:=
591:1
588:+
585:i
581:x
573:,
570:t
561:i
557:a
553:+
548:2
544:/
540:1
534:i
530:v
526:=
517:2
513:/
509:1
506:+
503:i
499:v
491:,
488:)
483:i
479:x
475:(
472:A
469:=
460:i
456:a
413:/
409:2
403:t
380:t
342:)
339:t
336:(
327:x
321:=
318:)
315:t
312:(
309:v
289:)
286:t
283:(
280:x
233:,
230:v
227:=
221:t
218:d
213:x
210:d
204:=
195:x
188:,
185:)
182:x
179:(
176:A
173:=
167:t
164:d
159:v
156:d
150:=
141:v
118:,
115:)
112:x
109:(
106:A
103:=
95:2
91:t
87:d
82:x
77:2
73:d
66:=
57:x
20:)
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