Stabilized Times Schemes for High Accurate Finite Differences Solutions of Nonlinear Parabolic Equations

TL;DR

This paper introduces a unified framework for Residual Smoothing Schemes (RSS) applied to high-order finite difference solutions of nonlinear parabolic equations, demonstrating their stability and efficiency in long-term simulations.

Contribution

It provides a unified theoretical framework, practical implementations, and stability analysis of RSS schemes for nonlinear parabolic problems using high-order finite differences.

Findings

RSS schemes are stable for nonlinear parabolic equations.

Numerical simulations show robustness in 2D Navier-Stokes problems.

The method reduces computational cost while maintaining accuracy.

Abstract

The Residual Smooting Scheme (RSS) have been introduced in \cite{AverbuchCohenIsraeli} as a backward Euler's method with a simplified implicit part for the solution of parabolic problems. RSS have stability properties comparable to those of semi-implicit schemes while giving possibilities for reducing the computational cost. A similar approach was introduced independently in \cite{BCostaPHD,CDGT} but from the Fourier point of view. We present here a unified framework for these schemes and propose practical implementations and extensions of the RSS schemes for the long time simulation of nonlinear parabolic problems when discretized by using high order finite differences compact schemes. Stability results are presented in the linear and the nonlinear case. Numerical simulations of 2D incompressible Navier-Stokes equations are given for illustrating the robustness of the method.