Volume 55, Number 3, May-June 2021
|Page(s)||1103 - 1131|
|Published online||08 June 2021|
Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation
College of Science, China University of Petroleum, Qingdao 266580, P.R. China
2 Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 11 April 2021
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.
Mathematics Subject Classification: 65M15 / 65M60
Key words: Local discontinuous Galerkin method / implicit-explicit time-marching scheme / stability; error estimate / compressible wormhole propagation
© EDP Sciences, SMAI 2021
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