General relaxation methods for initial-value problems with application to multistep schemes

Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or hig...

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Bibliographic Details
Published in:Numerische Mathematik 2020-12, Vol.146 (4), p.875-906
Main Authors: Ranocha, Hendrik, Lóczi, Lajos, Ketcheson, David I.
Format: Article
Language:English
Subjects:
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
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Summary:Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge–Kutta methods. We generalize this approach to multistep methods, including all general linear methods of order two or higher, and many other classes of schemes. We prove the existence of a valid relaxation parameter and high-order accuracy of the resulting method, in the context of general equations, including but not limited to conservative or dissipative systems. The theory is illustrated with several numerical examples.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01158-4