A Stable Explicit Scheme for Solving Inhomogeneous Constant Coefficients Differential Equation using Green's Function

TL;DR

This paper introduces a stable explicit numerical scheme for solving inhomogeneous linear PDEs with constant coefficients using Green's function, demonstrating improved stability over traditional finite difference methods.

Contribution

The paper presents a novel explicit scheme based on Green's function that remains stable regardless of time step size, unlike standard explicit finite difference methods.

Findings

The proposed method accurately matches exact solutions in simulations.

It remains stable for any time step, unlike explicit FDM.

Numerical results confirm the method's robustness and stability.

Abstract

A numerical explicit method to evaluates transient solutions of linear partial differential inhomogeneous equation with constant coefficients is proposed. A general form of the scheme for a specific linear inhomogeneous equation is shown. The method is applied to the wave equation and the diffuse equation and is investigated by simulating simple models. The numerical solutions of the proposed method show good agreement to the exact solutions. Comparing with explicit FDM, FDM shows the instability by the violation of CFL condition whereas the proposed method is always stable irrespective of any time step width.