Spectrally Accurate Energy-preserving Methods for the Numerical Solution of the "Good" Boussinesq Equation

TL;DR

This paper develops spectrally accurate, energy-preserving numerical methods for solving the 'good' Boussinesq equation, combining structure-preserving spatial discretization with energy-conserving time integrators to ensure accurate and stable solutions.

Contribution

The paper introduces a novel combination of Hamiltonian-preserving spatial discretization and energy-conserving Runge-Kutta methods for the 'good' Boussinesq equation.

Findings

Numerical tests confirm the effectiveness of the proposed methods.

The methods preserve the Hamiltonian structure over long simulations.

Spectral accuracy achieved in space and energy conservation in time.

Abstract

In this paper we study the geometric solution of the so called "good" Boussinesq equation. This goal is achieved by using a convenient space semi-discretization, able to preserve the corresponding Hamiltonian structure, then using energy-conserving Runge-Kutta methods in the HBVM class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.