Backward error analysis for variational discretisations of partial differential equations

TL;DR

This paper develops a backward error analysis framework for variational discretizations of PDEs, showing that symmetric solutions satisfy Hamiltonian modified equations with explicit structures, exemplified by rotating waves.

Contribution

It introduces a second-order Hamiltonian modified equation approach for symmetric solutions of discretized PDEs derived from variational principles, extending backward error analysis to this context.

Findings

Modified equations are Hamiltonian and have first-order Lagrangians.

Explicit computation for rotating travelling waves in nonlinear wave equations.

Symmetric solutions satisfy infinite-dimensional functional equations.

Abstract

In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.