Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

Abstract

In Klingenberg, Schnücke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L 2 -stability will be proven. Furthermore, an error estimate which provides the suboptimal (k + 1 2 ) convergence with respect to the L ∞ 0, T ; L 2 (Ω) -norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree k. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangularThe numerical flux function needs to satisfy certain properties. These properties are discussed in the Section 2.4.Finally, on the reference cell, the semi-discrete ALE-DG method appears as the following problem:Problem 1 (The semi-discrete ALE-DG method on the reference cell). Find a function(4.5)Proof. Since f (c) contains merely constant coefficients andthe integration by parts formula providesThus, we obtain the identity (4.5) by (4.6) and (4.7).Next, we assume that u * h = c solves the semi-discrete ALE-DG method Problem 1. Then, we obtain by (4.4) and (4.5)(4.8) 17The equation (4.8) and the ODE (2.12) are equivalent, since c is an arbitrary constant,is an arbitrary test function and the quantitiesare merely time-dependent. We note that the time evolution of the metric terms J K(t) needs to be respected in the time discretization of the semi-discrete ALE-DG method Problem 1. Therefore, we discretize the ODE (2.12) and (4.4) simultaneously by the same TVD-RK method. The stage solutions of the TVD-RK discretization for (2.12) will be used to update the metric terms in the TVD-RK discretization for (4.4).The fully-discrete ALE-DG method: First, the ODE (2.12) is discretized by a s-stage TVD-RK method:where K n+γ j := K (t n + γ j t) and ω n+γ j := ω (t n + γ j t). The stage solutions {J K n,i } s i=0 are used to update the metric terms in the TVD-RK discretization of (4.4). The Runge-Kutta method needs to solve the ODE (2.12) exact such thatwhere T (t n+1 ) is the regular mesh of simplices which has been used in the Section 2.1 to construct the time-dependent cells (2.3). We note that in a d-dimensional space a TVD-RK method with order greater than or equal to d is necessary to compute the metric ...

Positivity-Preserving Well-Balanced Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Methods for the Shallow Water Equations

“…In this section, we present the positivity-preserving well-balanced high-order ALE-DG methods for the shallow water equations which preserve the still water equilibrium. According to the previous work in [19,23], the ALE-DG methods can maintain almost all the mathematical properties of DG methods on static grids, such as conservation, geometric conservation law (GCL), entropy stability, and optimal error estimates. Our well-balanced ALE-DG methods for the still water equilibrium are mainly based on the GCL property on the moving mesh.…”

“…A two-step moving mesh DG scheme was presented in [42], where the well-balanced DG methods with hydrostatic reconstruction on static grids and a remapping were coupled. We will choose the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method developed by Klingenberg et al [19,23] in this paper, which maintains mathematical properties of the DG methods on static grids, such as conservation, geometric conservation law (GCL), entropy stability and high order accuracy. Thereinto, the GCL property of the grid deformation method is essential for the development of well-balanced grid deformation schemes for the shallow water equations.…”

“…Thereinto, the GCL property of the grid deformation method is essential for the development of well-balanced grid deformation schemes for the shallow water equations. It has been shown in [19] that the GCL property of the ALE-DG method can be satisfied for any time discretization methods with the order greater than or equal to the dimension. Thus the modified total variation diminishing Runge-Kutta (TVD-RK) methods has been introduced to guarantee the GCL property in each stage of Runge-Kutta methods for two or higher dimension cases.…”

High order weight‐adjusted discontinuous Galerkin methods for wave propagation on moving curved meshes

“…Recently, discontinuous Galerkin (DG) methods based on the ALE framework have been developed for their advantages of high order approximation on arbitrary unstructured meshes 12‐14 . A variety of DG spectral element methods (DG‐SEM) for wave propagation on moving domains can be found in References 3, 4, 12, and 15.…”

Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws

“…A lot of applications in engineering and physics require the approximation of conservation laws on time-dependent domains, e.g., domains with moving boundaries. For instance, moving mesh discontinuous Galerkin (DG) methods have been investigated in [4,24,43,50,51]. In particular, moving mesh discontinuous Galerkin spectral element methods (DGSEM) have been constructed and analyzed in [38,47,63].…”