论文标题
一种用于运输问题的欧拉 - 拉格朗日不连续的Galerkin方法及其在非线性动力学中的应用
An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics
论文作者
论文摘要
我们提出了一种新的Eulerian-Lagrangian（EL）不连续的Galerkin（DG）方法。该方法被设计为用于[J.中提出的线性对流问题的半拉格朗日（SL）DG方法的概括。科学。计算。 73：514-542，2017]，它是根据伴随问题制定的，并通过高度准确地跟踪特征曲线来追踪上游细胞。在SLDG方法中，取决于速度场，上游细胞可能具有任意形状。因此，需要更复杂的上游细胞的近似值才能获得高阶近似。例如，在旋转变形示例中，提出了二次曲线（QC）四边形以近似上游细胞，以获得三阶空间精度。在本文中，对于线性对流问题，我们提出了一种更通用的公式，称为ELDG方法。该方案是基于{\ em修饰}的伴随问题制定的，上游细胞始终是四边形，这避免了在SLDG算法中使用QC四边形的需求。 The newly proposed ELDG method can be viewed as a new general framework, in which both the classical Eulerian Runge-Kutta DG formulation and the SL DG formulation can fit in. 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.
We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017], which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. In the SLDG method, depending on the velocity field, upstream cells could be of arbitrary shape. Thus, a more sophisticated approximation to sides of the upstream cells is required to get high order approximation. For example, quadratic-curved (QC) quadrilaterals were proposed to approximate upstream cells for a third-order spatial accuracy in a swirling deformation example. In this paper, for linear advection problems, we propose a more general formulation, named the ELDG method. The scheme is formulated based on a {\em modified} adjoint problem for which the upstream cells are always quadrilaterals, which avoids the need to use QC quadrilaterals in the SLDG algorithm. The newly proposed ELDG method can be viewed as a new general framework, in which both the classical Eulerian Runge-Kutta DG formulation and the SL DG formulation can fit in. 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.
