Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media

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Department of Mathematical Sciences


In this paper, we construct bound-preserving interior penalty discontinuous Galerkin (IPDG) methods with a second-order implicit pressure explicit concentration (SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. Therefore, in all previous works, only the combination of forward and backward Euler time integrations was considered. In this paper, we propose a correction stage to compensate for the second-order accuracy in each time step. There are two main difficulties in constructing a second-order scheme. Firstly, in the concentration equation, correction of the diffusion term will cause anti-diffusion, leading to malfunction of the bound-preserving technique. We can deal with the velocity in the diffusion term explicitly to avoid correction of the diffusion term. Secondly, we need to ensure that the bound-preserving technique for the convection and source terms can be applied when the correction stage has been established. In fact, in the correction stage, the new approximation to the concentration can be chosen as the numerical solution in the previous stage, so the numerical cell averages are positivity-preserving. Moreover, we use the same correction technique for the pressure, so that the consistent flux pairs would guarantee the preservation of the upper bound 1 of the concentration. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. Moreover, the proposed method also yields much larger cfl number compared with first-order implicit pressure explicit concentration schemes. Moreover, the effectiveness of the bound-preserving technique will also be presented.

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Journal of Computational Physics