Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods
Abstract
When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is addressed by using an expensive, fully implicit Runge-Kutta method with high stage order or a specialized scheme satisfying additional order conditions. This work develops a flexible approach of augmenting an arbitrary Runge-Kutta method with a fully implicit method used to treat the forcing such as to maintain the classical order of the base scheme. Our methods and analyses are based on the general-structure additive Runge-Kutta framework. Numerical experiments using diagonally implicit, fully implicit, and even explicit Runge-Kutta methods confirm that the new approach eliminates order reduction for the class of problems under consideration, and the base methods achieve their theoretical orders of convergence.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2022
- DOI:
- arXiv:
- arXiv:2201.07940
- Bibcode:
- 2022arXiv220107940R
- Keywords:
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- Mathematics - Numerical Analysis;
- 65L04;
- 65L20