A Generalized Eulerian-Lagrangian Discontinuous Galerkin Method for Transport Problems
A Generalized Eulerian-Lagrangian Discontinuous Galerkin Method for Transport Problems
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DOI：
10.1016/j.jcp.2022.111160
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
Xue Hong;Jing-Mei Qiu
中科院分区：
文献类型：
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作者：
Xue Hong;Jing-Mei Qiu
提出了一种广义欧拉-拉格朗日间断Galerkin方法。该方法是Cai等人（2021）[5]提出的用于运输问题的欧拉-拉格朗日（EL）DG方法的推广，该方法沿着DG框架中的特征近似跟踪解决方案，允许超大的时间步长和稳定性。本文提出了一种求解变系数线性双曲型方程组的GEL DG方法，其中测试函数的伴随问题的速度场被冻结为常数。在本文中，在一个简化的标量设置，我们提出了GEL DG方法，通过冻结的速度场的伴随问题，并制定了半离散格式的时空区域划分的线性线近似的特征。采用龙格-库塔法得到全离散格式。我们进一步设计了通量限制器，以满足离散几何守恒律（DGCL）和最大值原理保持（MPP）的性质。对一维和二维线性迁移问题的数值结果表明了GEL DG方法的优良性质。这些包括高阶空间和时间精度，超大时间步长的稳定性，以及DGCL和MPP性能的满意度。
We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in Cai et al. (2021) [5], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties.