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An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics
Journal of Computational Physics ( IF 3.8 ) Pub Date : 2021-05-04 , DOI: 10.1016/j.jcp.2021.110392
Xiaofeng Cai , Jing-Mei Qiu , Yang Yang

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.



中文翻译:

输运问题的欧拉-拉格朗日间断Galerkin方法及其在非线性动力学中的应用

我们提出了一种新的欧拉-拉格朗日(EL)间断Galerkin(DG)方法,该方法通过为测试函数引入改进的伴随问题并通过在时空区域上进行PDE积分来实现,该时空区域由时间相关的线性函数近似特征划分。然后,可以通过新的通量项考虑修改后的伴随问题中的特性逼近中的误差,并且可以通过在线Runge-Kutta(RK)方法对其进行积分。ELDG框架被设计为半Lagrangian(SL)DG方法和经典Eulerian RK DG方法的推广,用于线性对流问题。它利用了两种配方的优势。在EL DG框架中,特性通过时间的线性函数来近似,因此,上游单元的形状在一般的二维问题中是四边形的。像SL DG方案一样,不需要二次弯曲的四边形来设计高于二阶的方案。另一方面,从我们的理论和数值研究中可以明显看出,经典的欧拉RK DG方法的时间步长约束得到了极大的缓解。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。提出了关于线性输运问题的数值结果,以及使用指数RK时间积分器的非线性Vlasov和不可压缩的Euler动力学,以证明ELDG方法的有效性。从我们的理论和数值研究中可以明显看出,经典的欧拉RK DG方法的时间步长约束得到了极大的缓解。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。提出了关于线性输运问题的数值结果,以及使用指数RK时间积分器的非线性Vlasov和不可压缩的Euler动力学,以证明ELDG方法的有效性。从我们的理论和数值研究中可以明显看出,经典的欧拉RK DG方法的时间步长约束得到了极大的缓解。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。提出了关于线性输运问题的数值结果,以及使用指数RK时间积分器的非线性Vlasov和不可压缩的Euler动力学,以证明ELDG方法的有效性。

更新日期:2021-05-11
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