Oscillation-free method for semilinear diffusion equations under noisy initial conditions

TL;DR

This paper introduces an oscillation-free numerical method for semilinear diffusion equations that effectively suppresses unwanted oscillations caused by noisy initial data, ensuring stable and accurate solutions.

Contribution

The authors develop a novel oscillation-free scheme using symmetric Strang splitting and a weighted implicit Euler approach for semilinear diffusion equations with noisy initial conditions.

Findings

The method successfully suppresses oscillations in numerical solutions.

It maintains stability and accuracy despite noisy initial data.

The approach outperforms traditional methods like Crank-Nicolson in noisy scenarios.

Abstract

Noise in initial conditions from measurement errors can create unwanted oscillations which propagate in numerical solutions. We present a technique of prohibiting such oscillation errors when solving initial-boundary-value problems of semilinear diffusion equations. Symmetric Strang splitting is applied to the equation for solving the linear diffusion and nonlinear remainder separately. An oscillation-free scheme is developed for overcoming any oscillatory behavior when numerically solving the linear diffusion portion. To demonstrate the ills of stable oscillations, we compare our method using a weighted implicit Euler scheme to the Crank-Nicolson method. The oscillation-free feature and stability of our method are analyzed through a local linearization. The accuracy of our oscillation-free method is proved and its usefulness is further verified through solving a Fisher-type equation where oscillation-free solutions are successfully produced in spite of random errors in the initial conditions.