A general framework for deriving integral preserving numerical methods for PDEs

TL;DR

This paper introduces a versatile framework for creating numerical methods that preserve invariants in PDEs, ensuring stability and accuracy for complex systems with polynomial nonlinearities.

Contribution

It presents a general, formalized procedure for deriving conservative, linearly implicit numerical integrators applicable to a wide class of PDEs with arbitrary variables and derivatives.

Findings

Second order convergence is formally proven.

Numerical experiments validate the effectiveness of the methods.

The framework is applicable to systems with polynomial nonlinearities.

Abstract

A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is developed for systems of partial differential equations with polynomial nonlinearities. The framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order. It is proved formally that second order convergence is obtained. The procedure is applied to a test case and numerical experiments are provided.