Excel实现四阶龙格库塔法(Runge-Kutta)解二阶常微分方程范例

h x解析解4RK解 y v, y'k1l1

0.010.000 1.000000 1.0000000.000000-2.0000000.000000 0.010.0100.9998990.999899-0.020252-2.050554-0.020252 0.010.0200.9995930.999593-0.041015-2.102231-0.041015 0.010.0300.9990770.999077-0.062300-2.155055-0.062300 0.010.040

0.010.050

0.010.060

0.010.070

0.010.080

0.010.090

0.010.100

0.010.110

0.010.120

0.010.130

0.010.140

0.010.150

0.010.160

0.010.170

0.010.180

0.010.190

0.010.200

0.010.210

0.010.220

0.010.230

0.010.240

0.010.250

0.010.2600.9154090.915409-0.726111-3.749153-0.726111 0.010.2700.9079590.907959-0.764046-3.838056-0.764046 0.010.2800.9001250.900125-0.802879-3.928888-0.802879 0.010.2900.8918990.891899-0.842630-4.021688-0.842630 0.010.3000.8832700.883270-0.883319-4.116498-0.883319 0.010.3100.8742290.874229-0.924967-4.213359-0.924967 0.010.3200.8647670.864767-0.967594-4.312315-0.967594 0.010.3300.8548740.854874-1.011220-4.413409-1.011220 0.010.3400.8445390.844539-1.055869-4.516686-1.055869 0.010.3500.8337530.833753-1.101562-4.622191-1.101562 0.010.3600.8225050.822505-1.148320-4.729970-1.148320 0.010.3700.8107830.810783-1.196169-4.840072-1.196169 0.010.3800.7985770.798577-1.245130-4.952544-1.245130 0.010.3900.7858770.785877-1.295228-5.067436-1.295228 0.010.4000.7726690.772669-1.346487-5.184798-1.346487 0.010.4100.7589430.758943-1.398932-5.304682-1.398932 0.010.4200.7446860.744686-1.452589-5.427139-1.452589 0.010.4300.7298870.729887-1.507484-5.552225-1.507484 0.010.4400.7145320.714532-1.563642-5.679992-1.563642 0.010.4500.6986100.698610-1.621092-5.810497-1.621092

0.010.4600.6821060.682106-1.679862-5.943797-1.679862 0.010.4700.6650080.665008-1.739978-6.079950-1.739978 0.010.4800.6473020.647302-1.801470-6.219015-1.801470 0.010.4900.6289740.628974-1.864368-6.361052-1.864368 0.010.5000.6100100.610010-1.928701-6.506124-1.928701

k2l2k3l3k4l4

-2.025000-0.010000-2.025275-0.010125-2.050556-0.020253 -2.076110-0.030505-2.076390-0.030632-2.102233-0.041016 -2.128354-0.051526-2.128641-0.051657-2.155057-0.062301 -2.181758-0.073076-2.182051-0.073209-2.209052-0.084121

-2.264244-0.106486

-2.320656-0.129410

-2.378315-0.152904

-2.437248-0.176980

-2.497480-0.201653

-2.559040-0.226934

-2.621954-0.252838

-2.686253-0.279378

-2.751964-0.306568

-2.819118-0.334422

-2.887745-0.362955

-2.957875-0.392182

-3.029541-0.422118

-3.102774-0.452778

-3.177607-0.484179

-3.254074-0.516336

-3.332209-0.549266

-3.412048-0.582986

-3.493625-0.617512

-3.576977-0.652864

-3.662141-0.689058

-3.749156-0.726113 -3.793129-0.744857-3.793601-0.745077-3.838059-0.764047 -3.882987-0.783236-3.883469-0.783461-3.928891-0.802881 -3.974792-0.822523-3.975284-0.822753-4.021691-0.842632 -4.068587-0.862739-4.069089-0.862973-4.116501-0.883321 -4.164412-0.903902-4.164925-0.904141-4.213363-0.924969 -4.262310-0.946034-4.262833-0.946278-4.312318-0.967595 -4.362324-0.989155-4.362858-0.989405-4.413413-1.011222 -4.464498-1.033287-4.465043-1.033543-4.516689-1.055871 -4.568877-1.078452-4.569434-1.078713-4.622194-1.101563 -4.675508-1.124672-4.676076-1.124939-4.729974-1.148322 -4.784437-1.171970-4.785017-1.172243-4.840076-1.196171 -4.895711-1.220369-4.896304-1.220647-4.952548-1.245132 -5.009381-1.269892-5.009986-1.270177-5.067440-1.295230 -5.125495-1.320565-5.126113-1.320855-5.184802-1.346489 -5.244105-1.372411-5.244736-1.372707-5.304686-1.398934 -5.365263-1.425455-5.365906-1.425758-5.427144-1.452591 -5.489021-1.479725-5.489677-1.480034-5.552229-1.507486 -5.615433-1.535245-5.616104-1.535561-5.679996-1.563645 -5.744555-1.592042-5.745240-1.592365-5.810502-1.621095 -5.876444-1.650145-5.877142-1.650475-5.943802-1.679864

-6.011156-1.709581-6.011869-1.709917-6.079955-1.739980 -6.148750-1.770378-6.149478-1.770722-6.219020-1.801473 -6.289286-1.832565-6.290029-1.832917-6.361058-1.864371 -6.432825-1.896173-6.433583-1.896532-6.506129-1.928704 -6.579429-1.961232-6.580203-1.961599-6.654298-1.994503