Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

In this paper,we shall establish the superconvergence properties of the Runge-Kutta dis-continuous Galerkin method for solving two-dimensional linear constant hyperbolic equa-tion,where the upwind-biased numerical flux is used.By suitably defining the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or different directions,we obtain the superconvergence results on the node averages,the numerical fluxes,the cell averages,the solution and the spatial derivatives.The superconvergence properties in space are pre-served as the semi-discrete method,and time discretization solely produces an optimal order error in time.Some numerical experiments also are given.

WENO Interpolation-Based and Upwind-Biased Free-Stream Preserving Nonlinear Schemes

For flow simulations with complex geometries and structured grids,it is preferred for high-order difference schemes to achieve high accuracy.In order to achieve this goal,the satisfaction of free-stream preservation is necessary to reduce the numerical error in the numerical evaluation of grid metrics.For the linear upwind schemes with flux splitting the free-stream preserving property has been achieved in the early study[Q.Li et al.,Commun.Comput.Phys.,22(2017),pp.64–94].In the current paper,new series of nonlinear upwind-biased schemes through WENO interpolation will be proposed.In the new nonlinear schemes,the shock-capturing capability on distorted grids is achieved,which is unavailable for the aforementioned linear upwind schemes.By the inclusion of fluxes on the midpoints,the nonlinearity in the scheme is obtained through WENO interpolations,and the upwind-biased construction is acquired by choosing relevant grid stencils.New third-and fifth-order nonlinear schemes are developed and tested.Discussions are made among proposed schemes,alternative formulations of WENO and hybrid WCNS,in which a general formulation of center scheme with midpoint and nodes employed is obtained as a byproduct.Through the numerical tests,the proposed schemes can achieve the designed orders of accuracy and free-stream preservation property.In 1-D Sod and Shu-Osher problems,the results are consistent with the theoretical predictions.In 2-D cases,the vortex preservation,supersonic inviscid flow around cylinder at M¥=4,Riemann problem,and shock-vortex interaction problems have been tested.More specifically,two types of grids are employed,i.e.,the uniform/smooth grids and the randomized/locally-randomized grids.All schemes can get satisfactory results in uniform/smooth grids.On the randomized grids,most schemes have accomplished computations with reasonable accuracy,except the failure of one third-order scheme in Riemann problem and shock-vortex interaction.Overall,the proposed nonlinear schemes have the capability to solve flow problems on badly deformed grids,and the schemes can be used in the engineering applications.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws:Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is an extension of the UFORCEmethod developed by Stecca,Siviglia and Toro[25],in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan.The proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic system.The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes.Finally,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.