Consistency and convergence of general linear multistep variable stepsize variable formula methods
Copyright policy )Consistency and convergence of general linear multistep variable stepsize variable formula methods
Linear multistep (LM) formulae are commonly used in the numerical solution of initial value problems of first order ordinary differential equations (ODE's). A rigorous theory for LM formulae, when these are implemented as constant stepsize constant formula methods, was developed after the publication of Dahlquist's classical paper [1] in 1956. After 1969 LM formulae have often been applied in practical codes as variable stepsize variable formula methods (VSVFM's). Therefore the development of a rigorous theory for LM formulae also in the case where these are used as VSVFM's is desirable. A formal definition of general LM VSVFM's is given in this paper. Then some theorems concerning the consistency and the convergence of general LM VSVFM's are formulated and proved. The results obtained in this paper can be extended for one-leg VSVFM's and for VSVFM's based on predictorcorrector schemes of different types.
convergence, consistency, one-leg methods, predictor-corrector schemes, zero-stability, Mesh generation, refinement, and adaptive methods for ordinary differential equations, multistep variable stepsize variable formula methods, order, Numerical methods for initial value problems involving ordinary differential equations, Stability and convergence of numerical methods for ordinary differential equations
convergence, consistency, one-leg methods, predictor-corrector schemes, zero-stability, Mesh generation, refinement, and adaptive methods for ordinary differential equations, multistep variable stepsize variable formula methods, order, Numerical methods for initial value problems involving ordinary differential equations, Stability and convergence of numerical methods for ordinary differential equations
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