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Problem

Given the IVP

y⁽¹⁾(t) = f( t, y(t) )
y(a) = y₀

approximate y(b).

Assumptions

The function f(t, y) should be continuous in both variables.

Tools

We will use Taylor series and iteration.

Initial Conditions

Choose a value of n ≥ 1. Set h = (b − a)/n.

Set t_i = a + ih for i = 0, 1, ..., n and let y_i be the approximation of y(t_i) for i = 1, ..., n.

Process

Multiple-step Euler's Method

For i = 1, 2, ..., n set

y_i = y_{i − 1} + h f(t_{i − 1}, y_{i − 1}).

Multiple-step Heun's Method

For i = 1, 2, ..., n calculate

K₀ = f(t_{i − 1}, y_{i − 1})
K₁ = f(t_i, y_{i − 1} + h K₀)

and set y_i = y_{i − 1} + h (K₀ + K₁)/2.

4th-order Runge Kutta

For i = 1, 2, ..., n calculate

K₀ = f(t_{i − 1}, y_{i − 1})
K₁ = f(t_{i − 1} + ½h, y_{i − 1} + ½h K₀)
K₂ = f(t_{i − 1} + ½h, y_{i − 1} + ½h K₁)
K₃ = f(t_i, y_{i − 1} + h K₂)

and set y_i = y_{i − 1} + h (K₀ + 2 K₁ + 2 K₂ + K₃)/6.

Approximating Intermediate Values

If a value of y(t) is required for a < t < b where t ≠ t_i for any i, choose the surrounding four points (three if t < t₁ or t > t_{n − 1}), find the interpolating polynomial and evaluate this polynomial at the point t. Alternatively, calculate the appropriate cubic spline where the derivatives at the end points are given by f(a, y₀) and f(b, y_n), respectively.

Topic 14.4: Multiple-Step Methods (HOWTO)