Lyapunov Computational Method for Two-Dimensional Boussinesq Equation

TL;DR

This paper introduces a numerical method using Lyapunov operators for solving the two-dimensional Boussinesq equation, combining order reduction and finite difference discretization, with proofs of stability, convergence, and numerical validation.

Contribution

It develops a novel Lyapunov-based numerical approach for the 2D Boussinesq equation, including stability and convergence analysis, and provides numerical validation.

Findings

Method is uniquely solvable

Proven stability and convergence

Numerical results validate theoretical analysis

Abstract

A numerical method is developed leading to Lyapunov 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 and analyzed for local truncation error for consistency. The stability is checked by using Lyapunov criterion and the convergence is studied. Some numerical implementations are provided at the end of the paper to validate the theoretical results.