Adaptive Energy Preserving Methods for Partial Differential Equations

TL;DR

This paper introduces a flexible numerical scheme that preserves energy integrals for PDEs on adaptive, non-uniform grids, applicable across various discretization methods, demonstrated on classical equations with promising results.

Contribution

It develops a novel energy-preserving scheme adaptable to different discretizations and grid adaptivity, enhancing numerical stability and accuracy for PDEs.

Findings

Successfully applied to Korteweg-de Vries and Sine-Gordon equations

Preserves first integrals on non-uniform, adaptive grids

Shows improved numerical stability and accuracy

Abstract

A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches, thereby also including finite element approaches. The schemes are then extended to accommodate $r$-, $h$- and $p$-adaptivity. The method is applied to the Korteweg-de Vries equation and the Sine-Gordon equation and results from numerical experiments are presented.