Lyapunov-Sylvester Computational Method for Two-Dimensional Boussinesq Equation

TL;DR

This paper introduces a novel numerical approach using Lyapunov-Sylvester operators to efficiently solve the two-dimensional Boussinesq equation, ensuring stability, convergence, and unique solvability.

Contribution

It develops a new computational method combining order reduction and finite difference discretization with Lyapunov-Sylvester operators for the Boussinesq equation.

Findings

Method is proven to be uniquely solvable, stable, and convergent.

Numerical examples validate theoretical stability and convergence.

Approach effectively approximates solutions of the 2D Boussinesq equation.

Abstract

A numerical method is developed leading to algebraic systems based on generalized Lyapunov-Sylvester operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite difference discretization. It is proved to be uniquely solvable, stable and convergent by using Lyapunov criterion and manipulating Lyapunov-Sylvester operators. Some numerical implementations are provided at the end of the paper to validate the theoretical results.