Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation

Abstract

In this paper, we apply two fully-discrete local discontinuous Galerkin (LDG) methods to the compressible wormhole propagation. We will prove the stability and error estimates of the schemes. Traditional LDG methods use the diffusion term to control of convection term to obtain the stability for some linear equations. However, the variables in wormhole propagation are coupled together and the whole system is highly nonlinear. Therefore, it is extremely difficult to obtain the stability for fully-discrete LDG methods. To fix this gap, we introduce a new auxiliary variable including both the convection and diffusion terms. Moreover, we also construct a special time integration for the porosity, leading to physically relevant numerical approximations and controllable growth rate of the porosity. With a reasonable growth rate, it is possible to handle the time level mismatch in the first-order fully discrete scheme and obtain the stability of the scheme. For the whole system, we will prove that under weak temporal-spatial conditions, the optimal error estimates for the pressure, velocity, porosity and concentration under different norms can be obtained. Numerical experiments are also given to verify the theoretical results.