8.3 Multi-step Methods and Stability Analysis

Multi-step methods for ODEs use information from previous steps to calculate the next solution point. This approach enhances efficiency compared to single-step methods, but requires multiple initial values to start the computation process.

These methods come in explicit and implicit forms, with Adams-Bashforth and Adams-Moulton being common examples. Stability analysis is crucial for understanding their behavior and choosing the right method for specific problems, especially stiff ones.

Multi-step Methods for ODEs

Concept of multi-step methods

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• Multi-step methods leverage information from previous steps to calculate next solution point enhancing efficiency compared to single-step methods
• Require multiple initial values to kickstart computation process
• General form of linear multi-step method expressed as $\sum_{j=0}^k \alpha_j y_{n+j} = h \sum_{j=0}^k \beta_j f_{n+j}$ where $\alpha_j$ and $\beta_j$ are coefficients, $h$ is step size
• Classified into explicit methods (use past information only) and implicit methods (solve equation at each step)

Implementation of Adams methods

• Adams-Bashforth methods (explicit) utilize polynomial interpolation of past derivative values
• General form: $y_{n+1} = y_n + h \sum_{j=0}^{k-1} b_j f_{n-j}$
• Common orders include:
  1. Two-step (second-order): $y_{n+1} = y_n + \frac{h}{2}(3f_n - f_{n-1})$
  2. Four-step (fourth-order): $y_{n+1} = y_n + \frac{h}{24}(55f_n - 59f_{n-1} + 37f_{n-2} - 9f_{n-3})$
• Adams-Moulton methods (implicit) incorporate current step in interpolation
• General form: $y_{n+1} = y_n + h \sum_{j=-1}^{k-1} b_j f_{n-j}$
• Common orders include:
  1. One-step (second-order): $y_{n+1} = y_n + \frac{h}{2}(f_{n+1} + f_n)$
  2. Three-step (fourth-order): $y_{n+1} = y_n + \frac{h}{24}(9f_{n+1} + 19f_n - 5f_{n-1} + f_{n-2})$
• Implementation requires single-step method (Runge-Kutta) to generate initial values
• Predictor-corrector pairs employ explicit method to predict, implicit to correct

Stability Analysis

Stability analysis with characteristic polynomials

• Characteristic polynomial derived from method's difference equation
• General form: $\rho(r) - h\lambda\sigma(r) = 0$ where $\rho(r)$ and $\sigma(r)$ are first and second characteristic polynomials
• Root condition dictates all roots of characteristic polynomial must have magnitude ≤ 1, roots with magnitude 1 must be simple
• Consistency achieved when $\rho(1) = 0$ and $\rho'(1) = \sigma(1)$
• Order of accuracy determined by comparing Taylor series expansion of method with true solution

Stability regions of multi-step methods

• Stability region encompasses complex values $h\lambda$ for which method remains stable
• Boundary locus method visualizes stability region in complex plane by setting $r = e^{i\theta}$ in characteristic equation
• A-stability achieved when stability region includes entire left half-plane (no explicit linear multi-step method can be A-stable)
• Adams-Bashforth methods generally exhibit smaller regions for higher orders
• Adams-Moulton methods typically boast larger regions than Adams-Bashforth
• Backward differentiation formulas (BDF) feature large regions, suitable for stiff problems
• Stiff problems necessitate methods with large stability regions, often favoring implicit methods for superior stability properties