MATH 175: NUMERICAL ANALYSIS II Lecturer: Jomar Fajardo Rabajante 2nd Sem AY 2012-2013 IMSP, UPLB

MULTISTEP METHODS These are ODE solvers that have order as high as the one-step methods but with less necessary function evaluations. Multistep methods can be generated using Taylor Series, or by integrating approximating interpolants (similar to what we did in Numerical Integration in Math 174).

Explicit Multistep Methods Adams-Bashforth Two-Step Method (Order 2) We start the method using the initial value and using one-step method of at least same order (e.g. Trapezoidal Method)

Explicit Multistep Methods Adams-Bashforth Three-Step Method (Order 3) Adams-Bashforth Four-Step Method (Order 4)

Implicit Multistep Methods Adams-Moulton Two-Step Method (Order 3) Use root-finding approach (if stiff), or the predictor-corrector approach (if non-stiff, e.g. partner Adams-Moulton and Adams-Bashforth methods) when dealing with the implicit part.

Implicit Multistep Methods Milne-Simpson Method (Order 4) Adams-Moulton Three-Step Method (Order 4)

Stability of Multistep Methods A general s-step method has the form

Stability of Multistep Methods DEFINITION: An s-step multistep method is stable if the roots of the characteristic polynomial are bounded by 1 in absolute value, and any roots of absolute value 1 are simple roots. A stable method for which 1 is the only root of absolute value 1 is called strongly stable; otherwise it is weakly stable.

Stability of Multistep Methods Example: Given the polynomial (from a general two-step formula) If the roots are 0 & 1, then strongly stable. If the roots are –2 & 1, then not stable. If the roots are –1 & 1, then weakly stable. If the root is 1 (multiplicity 2), then not stable.

Convergence of Multistep Methods THEOREM: A multistep method is convergent if and only if it is stable and consistent. Recall: A method is consistent if it has order at least 1.

Stability of Multistep Methods All discussed Adams-Bashforth methods are strongly stable. All discussed Adams-Moulton methods are strongly stable. Milne-Simpson Method is weakly stable (i.e. it is susceptible to error magnification)

Source: Society for Industrial and Applied Mathematics

Source: Society for Industrial and Applied Mathematics