Mimetic Finite Difference methods for Hamiltonian wave equations in 2D

TL;DR

This paper develops a numerical method combining Mimetic Finite Difference techniques with symplectic time integration to effectively solve Hamiltonian wave equations in two dimensions, preserving key physical properties.

Contribution

It introduces a novel combination of MFD spatial discretization with symplectic time integration for 2D Hamiltonian wave equations, with theoretical analysis and numerical validation.

Findings

The method accurately preserves the Hamiltonian structure.

Numerical simulations demonstrate stability and efficiency.

Theoretical analysis confirms the method's properties.

Abstract

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.