Options
An Auxiliary Inequality Based Method for Stabilization and Mesh-Adaptation of Steady and Time-Dependent Differential Equations
Date Issued
01-01-2021
Author(s)
Kumar, Vivek
Indian Institute of Technology, Madras
Abstract
We develop a general discrete inequality based on the entropy idea in hyperbolic conservation laws, and demonstrate that enforcing this auxiliary inequality can be utilized for a number of steady and time-dependent problems. We demonstrate that, for any existing central difference based method, addition of this auxiliary inequality at the discrete level, enables one to achieve several desired purposes. Firstly, the violation of the inequality allows us to determine unphysical regions in the numerical solution without any a-priori knowledge of the solution. Secondly, the sign of the discrete production also functions as an excellent indicator for mesh adaptation in several problems in general and singular perturbation problems in particular. Thirdly, the operator can be used to derive robust schemes for convection dominated problems. Most importantly, all these are achieved without any ad-hoc, user introduced, parameters. We provide a range of numerical results demonstrating the efficacy of the method and its applicability to both steady and time dependent problems.
Volume
1345