Compact finite-difference scheme for some Sobolev type equations with Dirichlet boundary conditions

TL;DR

This paper develops a stable, high-order compact finite difference scheme for Sobolev-type equations with Dirichlet boundary conditions, demonstrating accuracy, stability, and effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces a sixth-order compact finite difference method combined with explicit time stepping for Sobolev equations, with stability analysis and extensive numerical validation.

Findings

The scheme achieves sixth-order spatial accuracy.

The method is proven to be $L_2$-stable under certain conditions.

Numerical experiments confirm the scheme's effectiveness in various flow scenarios.

Abstract

This study aims to construct a stable, high-order compact finite difference method for solving Sobolev-type equations with Dirichlet boundary conditions in one-space dimension. Approximation of higher-order mixed derivatives in some specific Sobolev-type equations requires a bigger stencil information. One can approximate such derivatives on compact stencils, which are higher-order accurate and take less stencil information but are implicit and sparse. Spatial derivatives in this work are approximated using the sixth-order compact finite difference method (Compact6), while temporal derivatives are handled with the explicit forward Euler difference scheme. We examine the accuracy and convergence behavior of the proposed scheme. Using the von Neumann stability analysis, we establish $L_2-$stability theory for the linear case. We derive conditions under which fully discrete schemes are stable. Also, the amplification factor $\mathcal{C}(\theta)$ is analyzed to ensure the decay property over time. Real parts of $\mathcal{C}(\theta)$ lying on the negative real axis confirm the exponential decay of the solution. A series of numerical experiments were performed to verify the effectiveness of the proposed scheme. These tests include both one dimensional and two-dimensional cases of cases of advection-free and advection-diffusion flows. They also cover applications to the equal width equation, such as the propagation of a single solitary wave, interactions between two and three solitary waves, undular bore formation, and the Benjamin-Bona-Mahony-Burgers equation.