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

Abstract

The Residual Smoothing Scheme (RSS) have been introduced in Averbuch et al. (A fast and accurate multiscale scheme for parabolic equations, unpublished) as a backward Euler's method with a simplified implicit part for the solution of parabolic problems.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 Costa (Time marching techniques for the nonlinear Galerkin method, 1998), Costa et al. (SIAM J Sci Comput 23(1):46---65, 2001) 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.