Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media
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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DOI：
10.1016/j.jcp.2022.111240
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发表时间：
2022-04
期刊：
影响因子：
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通讯作者：
Wenjing Feng;Hui Guo;Yue Kang;Yang Yang-Yang
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文献类型：
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作者：
Wenjing Feng;Hui Guo;Yue Kang;Yang Yang-Yang
本文构造了二阶隐式压力显式浓度（SIPEC）时间推进的保界内罚间断Galerkin（IPDG）方法，用于求解二分量可压缩可混溶驱动耦合方程组。SIPEC方法是在传统的二阶强稳定性Runge-Kutta（SSP-RK 2）方法基础上的一个重要创新。其主要思想是隐式处理压力方程和显式处理浓度方程。然而，这种处理将导致一阶精确的方案。因此，在所有以前的工作中，只考虑向前和向后欧拉时间积分的组合。在本文中，我们提出了一个校正阶段，以补偿在每个时间步的二阶精度。构造二阶格式有两个主要困难。首先，在浓度方程中，对扩散项的修正会引起反扩散，导致边界保持技术失效。我们可以显式地处理扩散项中的速度，以避免对扩散项的校正。其次，我们需要确保对流项和源项的边界保持技术在修正阶段建立后可以应用。事实上，在校正阶段，可以选择浓度的新近似值作为前一阶段的数值解，因此数值单元平均值是保正的。此外，我们使用相同的校正技术的压力，使一致的通量对将保证浓度的上限1的保存。数值实验表明，当浓度方程中扩散系数D较小时，与显式格式相比，该格式可以显著降低计算量.此外，与一阶隐式压力显式浓度格式相比，该方法也产生了更大的cfl数。此外，还将介绍边界保持技术的有效性。
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 coefficientDis 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.