An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics
An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics
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DOI：
10.1016/j.jcp.2021.110392
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Xiaofeng Cai;Jing-Mei Qiu;Yang Yang-Yang
中科院分区：
文献类型：
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作者：
Xiaofeng Cai;Jing-Mei Qiu;Yang Yang-Yang
我们提出了一种新的欧拉-拉格朗日 (EL) 不连续伽辽金 (DG) 方法，该方法通过引入测试函数的改进伴随问题并通过在由近似特征的时间相关线性函数划分的时空区域上执行偏微分方程的积分来制定。然后，可以通过新的通量项考虑修正伴随问题中的特征近似中产生的误差，并且可以通过线性龙格-库塔（RK）方法进行积分。 ELDG 框架被设计为线性平流问题的半拉格朗日 (SL) DG 方法和经典欧拉 RK DG 方法的推广。它利用了两种配方的优点。在EL DG框架中，特性通过时间上的线性函数来近似，因此上游单元的形状在一般二维问题中是四边形。设计高于二阶的方案（如 SL DG 方案）不需要二次曲线四边形。另一方面，经典欧拉 RK DG 方法的时间步长约束大大减轻，这一点从我们的理论和数值研究中可以明显看出。观察到所提出的 EL DG 方法与任意拉格朗日-欧拉 (ALE) DG 的联系。线性输运问题以及使用指数 RK 时间积分器的非线性 Vlasov 和不可压缩欧拉动力学的数值结果证明了 ELDG 方法的有效性。
We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.