Multistep Methods

Numerical Analysis · Axiom Academy

LESSON Multistep Methods for ODEs Leveraging the history of computed solutions to achieve higher efficiency: using multiple previous values to advance numerical solutions Single-step methods (Euler, RK4) compute y_ n+1 using only information from the current step y_n. Each step is independent of the history. Multistep methods leverage the "memory" of previous computations. They use multiple past values y_ n , y_ n-1 , y_ n-2 , ... to compute y_ n+1 . 2. Linear Multistep Method Form The general k-step linear multistep method takes the form: where the coefficients alpha_j and beta_j define the specific method The method uses the last k+1 values of y and f to compute the next value. Different choices of alpha and beta coefficients yield different methods with varying accuracy and stability properties. 3. Explicit vs Implicit Methods The key distinction lies in whether beta_k (the coefficient of f_ n+1 ) is zero: beta_k = 0: No f_ n+1 term, direct computation beta_k != 0: Requires solving for y_ n+1 4. Derivation via Interpolation Multistep methods are derived by fitting a polynomial through previous points and then integrating: The key idea: approximate f(x, y(x)) by a polynomial P(x) that passes through the known points (x_ n-j , f_ n-j ), then integrate P(x) exactly. 1. Fit a line through (x_ n-1 , f_ n-1 ) and (x_n, f_n) 2. Integrate this line from x_n to x_ n+1 3. Result: y_ n+1 = y_n + h/2(3f_n - f_ n-1 )

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