Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations
Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations
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DOI：
10.1016/j.camwa.2022.05.022
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发表时间：
2021-10
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通讯作者：
Thi-Thao-Phuong Hoang
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作者：
Thi-Thao-Phuong Hoang
本文研究非均质多孔介质中输运问题的数值解。采用混合杂交有限元方法得到了线性对流扩散方程的半离散时间连续格式，其中通量变量同时代表对流和扩散通量，对流项的离散采用杂交产生的拉格朗日乘子.基于时间域和非重叠区域分解，我们提出了两种求解半离散问题的隐式局部时间步方法。第一种方法使用时间相关的Steklov-Poincar\'e型算子，第二种方法使用具有Robin传输条件的优化的施瓦茨波形松弛（OSWR）。对于每一种方法，我们制定了一个空间-时间界面的问题，迭代求解。每次迭代都涉及在时间上独立和全局地解决子域问题;因此，可以在子域中使用不同的时间步长。证明了非匹配网格下全离散OSWR算法的收敛性。不同的Pecl\'et数和不连续系数的问题，包括一个原型的模拟核废料的地下存储的数值结果，来说明所提出的本地时间步长方法的性能。
This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincar\'e type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Pecl\'et numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.