Galerkin Method For Conservation Laws Research Articles

A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes

The objective of the present work is to develop a new numerical framework for simulations involving deformable domains, in the specific context of high-order meshes consistent with Computer-Aided Design (CAD) representations. Thus, the proposed approach combines ideas from isogeometric analysis, able to handle exactly CAD-based geometries, and Discontinuous Galerkin (DG) methods with an Arbitrary Lagrangian-Eulerian (ALE) formulation, able to solve complex problems with moving grids. The resulting approach is a DG method based on rational Bézier elements, that can be easily constructed from Non-Uniform Rational B-Splines (NURBS), formulated in a general ALE setting. We focus here on applications in two-dimensional compressible flows, but the method could be applied to other models as well. Two verification exercises are conducted, to assess rigorously the properties of the method and the convergence rates for representations up to sixth order. Finally, three problems are analysed in depth, involving compressible Euler and Navier-Stokes equations, for an oscillating cylinder and a pitching airfoil. In particular, the convergence of flow characteristics is investigated, as well as the impact of using curved boundaries in the context of deformable domains.

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

In Klingenberg, Schn\"ucke 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 L2-stability will be proven. Furthermore, an error estimate which provides the suboptimal (k+1/2) convergence with respect to the L-infinity-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 triangular meshes.

A reconstructed central discontinuous Galerkin method for conservation laws

Abstract Central discontinuous Galerkin (CDG) methods are a family of high order numerical methods, which evolve two sets of numerical solutions defined on overlapping meshes and do not employ the numerical fluxes at element interfaces as in discontinuous Galerkin methods. However, evolving two sets of numerical solutions makes the CDG method relatively time-consuming. In this paper, we present a reconstructed central discontinuous Galerkin (RCDG) method for conservation laws. In this scheme, we reconstruct an approximate solution by projecting the numerical solution defined on the primal mesh into the approximate space defined on the dual mesh in the L 2 sense. As a consequence, the reconstructed approximate solution plays a role as an alternative for the numerical solution defined on the dual mesh in the CDG method. Numerical examples demonstrate that the RCDG method is not only simpler and easier to implement than the CDG method but also reduces the computational cost considerably without sacrificing the high order accuracy of the CDG method.

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

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 12) convergence for monotone fluxes, and optimal (k + 1) convergence for an upwind flux, when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.

A discontinuous Galerkin method for two-layer shallow water equations

In this paper, we study a discontinuous Galerkin method to approximate solutions of the two-layer shallow water equations on non-flat topography. The layers can be formed in the shallow water model based on the vertical variation of water density which in general depends on the water temperature and salinity. For a water body with equal density the model reduces to the canonical single-layer shallow water equations. Thus, for a model with equal density on flat bottom, the method is equivalent to the discontinuous Galerkin method for conservation laws. The considered method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hp-adaptivity. Several numerical results illustrate the performance of the method and confirm its capability to solve two-layer shallow water flows including tidal conditions on the water free-surface and bed frictions on the bottom topography.

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal ( k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.

A discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations

In this paper, we propose a new discontinuous Galerkin finite element method to solve the Hamilton–Jacobi equations. Unlike the discontinuous Galerkin method of [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690.] which applies the discontinuous Galerkin framework on the conservation law system satisfied by the derivatives of the solution, the method in this paper applies directly to the solution of the Hamilton–Jacobi equations. For the linear case, this method is equivalent to the traditional discontinuous Galerkin method for conservation laws with source terms. Thus, stability and error estimates are straightforward. For the nonlinear convex Hamiltonians, numerical experiments demonstrate that the method is stable and provides the optimal ( k + 1)th order of accuracy for smooth solutions when using piecewise kth degree polynomials. Singularities in derivatives can also be resolved sharply if the entropy condition is not violated. Special treatment is needed for the entropy violating cases. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme.

A Local Discontinuous Galerkin Method for KdV Type Equations

In this paper we develop a local discontinuous Galerkin method for solving KdV type equations containing third derivative terms in one and two space dimensions. The method is based on the framework of the discontinuous Galerkin method for conservation laws and the local discontinuous Galerkin method for viscous equations containing second derivatives; however, the guiding principle for intercell fluxes and nonlinear stability is new. We prove L2 stability and a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type in both one and multiple space dimensions, and we give an error estimate for the linear cases in the one-dimensional case. The stability result holds in the limit case when the coefficients to the third derivative terms vanish; hence the method is especially suitable for problems which are "convection dominated," i.e., those with small second and third derivative terms. Numerical examples are shown to illustrate the capability of this method. The method has the usual advantage of local discontinuous Galerkin methods, namely, it is extremely local and hence efficient for parallel implementations and easy for h-p adaptivity.

The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems

This is the fifth paper in a series in which we construct and study the so-called Runge–Kutta discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two-dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles on the quality of the approximation.