Optimal parameters for numerical solvers of PDEs

TL;DR

This paper presents an adaptive procedure to optimize parameters in numerical PDE solvers, enhancing accuracy and efficiency while preserving conservation properties, demonstrated on Korteweg-de Vries and nonlinear heat equations.

Contribution

Introduces a novel adaptive parameter optimization method for numerical PDE schemes that improves accuracy and efficiency while maintaining conservation properties.

Findings

Enhanced accuracy in numerical solutions.

Reduced computational overheads.

Improved efficiency over existing methods.

Abstract

In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.