Two-phase miscible flow in porous media is described by a coupled system of nonlinear partial differential equations. In this paper, we develop and analyze the fully discrete local discontinuous Galerkin method coupled with the implicit-explicit Runge Kutta (IMEX-RK) method for the problems. Second-order time discretization scheme in two adjacent time layer can be obtained with the IMEX-RK method. For the temporal-spatial discretization schemes, we give the optimal convergence analysis for both the pressure and concentration in $L^2$-norm. We present several numerical examples for two-phase miscible flow problem to demonstrate the effectiveness and robustness of our method.

多孔介质中的两相混相流动由非线性偏微分方程的耦合系统描述。在本文中，我们针对这些问题开发并分析了完全离散的局部不连续伽辽金方法与隐显式龙格库塔 (IMEX-RK) 方法相结合的方法。IMEX-RK 方法可以得到两个相邻时间层的二阶时间离散化方案。对于时空离散化方案，我们给出了压力和浓度的最优收敛分析${升}^2$-规范。我们提出了两相混相流动问题的几个数值例子，以证明我们方法的有效性和鲁棒性。