Numerical analysis of a time relaxation finite difference method for the heat equation

Abstract

In this study, we first consider the time-relaxation model, which consists of adding the term $\kappa \left( u-\overline{u}\right) $ to the heat equation. Then, an explicit discretization scheme for the model is introduced to find the finite difference solutions. We first obtain the solutions by using the scheme and then investigate the method’s consistency, stability, and convergence properties. We prove that the method is consistent and unconditionally stable for any given value of $r$ and appropriate values of $\kappa$ and $\delta$. As a result, the method obtained by adding the time relaxation term to the first-order finite-difference explicit method behaves like the second-order implicit method. Finally, we apply the method to some test examples.

Keywords

• Heat equation
• time-relaxation model
• differential filter
• finite difference method

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