Stabilized Times Schemes for High Accurate Finite Differences Solutions of Nonlinear Parabolic Equations
For computational scientists solving parabolic PDEs, this work provides a computationally cheaper alternative to semi-implicit schemes with comparable stability, though it is an incremental extension of existing methods.
The paper presents a unified framework for Residual Smoothing Schemes (RSS) for solving nonlinear parabolic equations with high-order finite differences, demonstrating stability in linear and nonlinear cases and robustness via 2D Navier-Stokes simulations.
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.