A Time Variable Filter Approach for the Fluid-Fluid Interaction

Abstract

This paper analyzes a time variable filter approach for the nonlinear fluid-fluid interaction problem. The method applies a time variable filter to improve numerical solution of a variable time step Euler scheme with explicit second-order extrapolation treatment for nonlinear convection and interface terms. Compared with classical second-order method in time, the proposed approach improves time accuracy from the first order to the second order by adding several lines to the code of variable time step Euler scheme. Theoretically, we prove the unconditional energy stability, local $H^1$ stability and error estimates. Numerically, some numerical experiments are provided to test the theoretical results, which illustrate the accuracy and efficiency of the presented method.