RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes

In this paper,a new multi-resolution weighted essentially non-oscillatory(MR-WENO)limiter for high-order local discontinuous Galerkin(LDG)method is designed for solving Navier-Stokes equations on triangular meshes.This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes.Such new limiter uses information of the LDG solution essentially only within the troubled cell itself,to build a sequence of hierarchical L^(2)projection polynomials from zeroth degree to the highest degree of the LDGmethod.As an example,a third-order LDGmethod with associated same orderMR-WENO limiter has been developed in this paper,which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities.The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one.This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell.This MR-WENO limiter is very simple to construct,and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes.Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes.Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

The arbitrary Lagrangian-Eulerian(ALE)method is widely used in the field of compressible multi-material and multi-phase flow problems.In order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin(DG)discretizations,we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this paper.This remapping method is based on the ALE-DG method developed by Klingenberg et al.[17,18]to solve the trivial equation∂u/∂t=0 on a moving mesh,which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=T.An appropriate selection of the final pseudo-time T can always satisfy the relatively mild smoothness requirement(Lipschitz continuity)on the mesh movement velocity,which guarantees the high order accuracy of the remapping procedure.We use a multi-resolution weighted essentially non-oscillatory(WENO)limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions.We further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy.Numerical experiments are provided to illustrate the high order accuracy,essentially non-oscillatory performance and positivity-preserving of our remapping algorithm.In addition,the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one-and two-dimensional Euler equations.

LWDG Method for a Multi-Class Traffic Flow Model on an Inhomogeneous Highway

In this paper,we apply the discontinuous Galerkin method with LaxWendroff type time discretizations(LWDG)using the weighted essentially nonoscillatory(WENO)limiter to solve a multi-class traffic flow model for an inhomogeneous highway.This model is a kind of hyperbolic conservation law with spatially varying fluxes.The numerical scheme is based on a modified equivalent system which is written as a“standard”hyperbolic conservation form.Numerical experiments for both the Riemann problem and the traffic signal control problem are presented to show the effectiveness of the method.