Numerical Resolution near t = 0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

TL;DR

This paper investigates the impact of corner singularities caused by initial-boundary data incompatibilities on numerical solutions of 1D nonlinear evolution equations and proposes two methods to improve accuracy.

Contribution

It introduces two novel remedy procedures that effectively mitigate corner singularity effects in numerical schemes for 1D nonlinear evolution equations.

Findings

Remedy procedures improve numerical accuracy near t=0.

Applications demonstrated on Burgers and reaction-diffusion equations.

Methods applicable to various nonlinear diffusion equations.

Abstract

The incompatibilities between the initial and boundary data will cause singularities at the time-space corners, which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions. We study the corner singularity issue for nonlinear evolution equations in 1D, and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use. Applications of the remedy procedures to the 1D viscous Burgers equation, and to the 1D nonlinear reaction-diffusion equation are presented. The remedy procedures are applicable to other nonlinear diffusion equations as well.