Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

Klingenberg, Christian; Schnücke, Gero; Xia, Yun‐Jie

doi:10.1090/mcom/3126

Cited by 42 publications

(24 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Zhang

Xia

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

In this paper, we develop well-balanced arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the shallow water equations, which preserve not only the still water equilibrium but also the moving water equilibrium. Based on the time-dependent linear affine mapping, the ALE-DG method for conservation laws maintains almost all mathematical properties of DG methods on static grids, such as conservation, geometric conservation law (GCL), entropy stability. The main difficulty to obtain the well-balanced property of the ALE-DG method for shallow water equations is that the grid movement and usual time discretization may destroy the equilibrium. By adopting the GCL preserving Runge-Kutta methods and the techniques of well-balanced DG schemes on static grids, we successfully construct the high order well-balanced ALE-DG schemes for the shallow water equations with still and moving water equilibria. Meanwhile, the ALE-DG schemes can also preserve the positivity property at the same time. Numerical experiments in different circumstances are provided to illustrate the well-balanced property, positivity preservation and high order accuracy of these schemes.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Zhang

Xia

2021

J Sci Comput

Self Cite

View full text Add to dashboard Cite

show abstract

“…The computational domain is [0, 1] and the final time is T = 0.15. The density using 100 cells is shown in figure (22) with static and moving meshes. The mesh motion does not significantly improve the solution compared to the static mesh case since the solution is smooth.…”

Section: Problemmentioning

confidence: 99%

Single-Step Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

Badwaik

Chandrashekar

Klingenberg

2020

Commun. Appl. Math. Comput.

Self Cite

View full text Add to dashboard Cite

We propose an explicit, single step discontinuous Galerkin (DG) method on moving grids using the arbitrary Lagrangian-Eulerian (ALE) approach for one dimensional Euler equations. The grid is moved with the local fluid velocity modified by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

show abstract

“…Ciarlet [4]). In particular, a one dimensional proof is given in [14] and in [15] two dimensional error analysis for a semi-discrete ALE-DG method to solve the Hamilton-Jacobi equations is given. In addition, error analysis for the fully-discrete ALE-DG methods is given in [43].…”

Section: The Numerical Flux Functionmentioning

confidence: 99%

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

Fu¹,

Schnücke²,

Xia³

2019

Math. Comp.

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

Cited by 42 publications

References 33 publications

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

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

Single-Step Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

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

Contact Info

Product

Resources

About