Galerkin Scheme For Conservation Laws Research Articles

A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case

The concept of multiresolution-based adaptive DG schemes for non-linear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in (Math Comp, doi: 10.1090/S0025-5718-2013-02732-9 , 2013). The key idea is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and non-linear scalar conservation laws.

Adaptive multiresolution discontinuous Galerkin schemes for conservation laws

A multiresolution-based adaptation concept is proposed that aims at accelerating Discontinuous Galerkin schemes applied to nonlinear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels, that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. A central mathematical problem addressed in this work is then to show that the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh. Numerical comparisons demonstrate the efficiency of the concept and provide reliable estimates of the actual error in the numerical solution.

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091---3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918---8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.

A Discontinuous Galerkin Scheme Based on a Space–Time Expansion. I. Inviscid Compressible Flow in One Space Dimension

In this paper, we propose an explicit discontinuous Galerkin scheme for conservation laws which is of arbitrary order of accuracy in space and time. The basic idea is to use a Taylor expansion in space and time to define a space---time polynomial in each space---time element. The space derivatives are given by the approximate solution at the old time level, the time derivatives and the mixed space---time derivatives are computed from these space derivatives using the so-called Cauchy---Kovalevskaya procedure. The space---time volume integral is approximated by Gauss quadrature with values at the space---time Gaussian points obtained from the Taylor expansion. The flux in the surface integral is approximated by a numerical flux with arguments given by the Taylor expansions from the left and from the right-hand side of the element interface. The locality of the presented method together with the space---time expansion gives the attractive feature that the time steps may be different in each grid cell. Hence, we drop the common global time levels and propose that every grid zone runs with its own time step which is determined by the local stability restriction. In spite of the local time steps the scheme is locally conservative, fully explicit, and arbitrary order accurate in space and time for transient calculations. Numerical results are shown for the one-dimensional Euler equations with orders of accuracy one up to six in space and time.

Two‐Dimensional Surges and Shocks in Open Channels

The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow, although the complex geometry of most practical problems makes the use of finite elements very desirable. A variance of the Galerkin scheme for conservation laws in two‐dimensional, nearly horizontal flow, which exhibits a remarkable shock‐capturing ability, is presented. The parasitic waves in the vicinity of the discontinuity commonly present in the Gelerkin solution are selectively dissipated, and, in fact, the sharpness of the front is improved by the addition of the dissipation mechanism. The method is based on discontinuous weighting functions which introduce upwind effects in the solution while maintaining central difference accuracy. No arbitrary parameters are needed because the dissipation level is selected analytically. No higher order derivatives appear in the governing equations which simplifies the construction and execution of the scheme. Results are presented for several flow situations that give rise to spontaneous formation of surge and shock waves in two space dimensions. The accuracy of computation is verified by comparison with analytical solutions and by continuous monitoring of the conservation properties of the model.