Simple bespoke preservation of two conservation laws

• Published in: IMA journal of numerical analysis Vol. 40; no. 2; pp. 1294 - 1329
• Main Authors: Frasca-Caccia, Gianluca, Hydon, Peter Ellsworth
• Format: Journal Article
• Language: English
• Published: Oxford University Press 24-04-2020
• Subjects: finite difference methods; KdV equation; porous medium equation; nonlinear heat equation; discrete conservation laws
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/dry087

Abstract
Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.