An efficient variational multiscale element free Galerkin method based on adaptive TR-AB2 scheme for convection–diffusion equations

Abstract

The variational multiscale element-free Galerkin (VMEFG) method can effectively improve stability when solving convection-dominated problems. However, for time-dependent problems, the VMEFG method has a large amount of calculation. In this paper, an efficient VMEFG method for solving unsteady convection-dominated convection–diffusion equations is proposed. The trapezoidal rule (TR) with adaptive time step is used for time integration, and the explicit second-order Adams–Bashforth (AB2) method is used to control the error. The main idea is to allow the time step to change along the direction of the solution, so that the amount of calculation under the specified accuracy requirements can be minimized. We further exemplified the effectiveness and accuracy of the given method in several examples. The numerical results show that the VMEFG method combined with TR-AB2 method (VMEFG-TR-AB2) can effectively increase computing efficiency while maintaining the calculation accuracy for the convection-dominated problem.