Solving Initial-Value Problem for ODE
Initial value problem：
1. Single step methods：
Runge-Kutta methods：
（1） Euler’s method：
,
, for each
（2） Midpoint method：
,
, for each
（3） Modified Euler method：
,
, for each
（4） Heun’s method：
,
, for each
（5） Classic Runge-Kutta method：
for each
（6） Runge-Kutta Fehlberg method：
This technique consists of using a Runge-Kutta method with local truncation error of order five,
,
to estimate the local error in a Runge-Kutta method of order four,
,
where
,
,
,
,
,
.
An advantage to this method is that only six evaluations of are required per step.
2. Linear multistep methods：
Adams-Bashforth methods：
（1） One-step: Euler’s method.
（2） Two-step:
,
, for each
（3） Three-step:
,
, for each
（4） Four-step:
,
for each
Adams-Moulton methods：
（1） One-step: Trapezoidal method,
,
, for each
（2） Two-step:
,
, for each
（3） Three-step:
,
, for each
Adams Fourth-Order Predictor-Corrector：
,
.
Adams Variable-Step-Size Fourth-Order Predictor-Corrector：
,
,
with an estimate of the local truncation error of the corrector step:
,
followed by a process of choosing and a criterion of changing when necessary. A change in step size for a multistep method is more costly in terms of function evaluations than for a one-step method, since new equally spaced starting values must be computed. So we tend to do this conservatively.
Milne’s method：
.
Simpson’s method：
.
Milne-Simpson Predictor-Corrector：
,
.
Note: However, Milne, Simpson, Milne-Simpson Predictor-Corrector methods are all inherently A-unstable and therefore are of limited use.
BDF（Backward Differential Formula）：
（1） One-step: implicit Euler’s method,
,
.
（2） Two-step:
,
.
（3） Three-step:
,
.
（4） Four-step:
.
（5） Five-step:
,
（6） Six-step:
,
3. MATLAB ODE suite：
Non-Stiff：ode23, ode45, ode113.
Stiff：ode15s, ode23s.
For details read 【2】.
References:
【1】
R. L. Burden and J. D. Faires, Numerical Analysis, PWS,
【2】 L. F. Shampine and M. W. Reichelt, “The Matlab ODE Suite,” Siam Journal on Scientific Computing, Volume 18, Number 1, pp1-22, 1997.