Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation
Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time marching for simulating wormhole propagation
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DOI：
10.1051/m2an/2021020
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
Hui Guo;Rui-yu Jia;Lulu Tian;Yang Yang-Yang
中科院分区：
文献类型：
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作者：
Hui Guo;Rui-yu Jia;Lulu Tian;Yang Yang-Yang
本文将两种全离散的局部间断Galerkin（LDG）方法应用于可压缩虫孔传播问题。我们将证明该方案的稳定性和误差估计。传统的LDG方法利用扩散项来控制对流项，以获得某些线性方程的稳定性。然而，虫洞传播中的变量是耦合在一起的，整个系统是高度非线性的。因此，这是非常困难的，以获得全离散LDG方法的稳定性。为了解决这个差距，我们引入了一个新的辅助变量，包括对流和扩散项。此外，我们还构建了一个特殊的时间积分的孔隙度，导致物理相关的数值近似和可控的孔隙度的增长率。在合理的增长率下，可以处理一阶全离散格式的时间失配，并保证格式的稳定性。对于整个系统，我们将证明在较弱的时空条件下，可以得到压力、速度、孔隙度和浓度在不同范数下的最优误差估计。数值实验也验证了理论结果。
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.