A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Assessment of shock capturing schemes for discontinuous Galerkin method

This paper carries out systematical investigations on the performance of several typical shock-capturing schemes for the discontinuous Galerkin （DG） method, including the total variation bounded （TVB） limiter and three artificial diffusivity schemes （the basis function-based （BF） scheme, the face residual-based （FR） scheme, and the element residual-based （ER） scheme）. Shock-dominated flows （the Sod problem, the Shu- Osher problem, the double Mach reflection problem, and the transonic NACA0012 flow） are considered, addressing the issues of accuracy, non-oscillatory property, dependence on user-specified constants, resolution of discontinuities, and capability for steady solutions. Numerical results indicate that the TVB limiter is more efficient and robust, while the artificial diffusivity schemes are able to preserve small-scale flow structures better. In high order cases, the artificial diffusivity schemes have demonstrated superior performance over the TVB limiter.

ON THE DISCRETE MAXIMUM PRINCIPLE FOR THE LOCAL PROJECTION SCHEME WITH SHOCK CAPTURING

It is a well known fact that finite element solutions of convection dominated problems can exhibit spurious oscillations in the vicinity of boundary layers. One way to overcome this numerical instability is to use schemes that satisfy the discrete maximum principle. There are monotone methods for piecewise linear elements on simplices based on the up- wind techniques or artificial diffusion. In order to satisfy the discrete maximum principle for the local projection scheme, we add an edge oriented shock capturing term to the bilinear form. The analysis of the proposed stabilisation method is complemented with numerical examples in 2D.

Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

Construction of Conservative Numerical Fluxes for the Entropy Split Method

The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts(SBP)difference boundary closure of(Gerritsen and Olsson in J Comput Phys 129:245-262,1996;Olsson and Oliger in RIACS Tech Rep 94.01,1994;Yee et al.in J Comp Phys 162:33-81,2000).Sj?green and Yee(J Sci Comput)recently proved that the entropy split method is entropy conservative and stable.Stand-ard high-order spatial central differencing as well as high order central spatial dispersion relation preserving(DRP)spatial differencing is part of the entropy stable split methodol-ogy framework.The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives.Due to the construction,this conservative numerical flux requires higher oper-ations count and is less stable than the original semi-conservative split method.However,the Tadmor entropy conservative(EC)method(Tadmor in Acta Numerica 12:451-512,2003)of the same order requires more operations count than the new construction.Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative,a modified nonlinear filter approach of(Yee et al.in J Comput Phys 150:199-238,1999,J Comp Phys 162:3381,2000;Yee and Sj?green in J Comput Phys 225:910934,2007,High Order Filter Methods for Wide Range of Compressible flow Speeds.Proceedings of the ICOSAHOM09,June 22-26,Trondheim,Norway,2009)is proposed in conjunction with the entropy split method as the base method for problems containing shock waves.Long-time integration of 2D and 3D test cases is included to show the com-parison of these new approaches.

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially nonoscillatory(WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifthorder WENO-JS(WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable.The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On nonuniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime,the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.

A Straightforward hp-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods

In this paper,high-order Discontinuous Galerkin(DG)method is used to solve the two-dimensional Euler equations.A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities.Numerical tests show that the shocks can be captured within one element even on very coarse grids.The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions.In order to obtain better shock resolution,a straightforward hp-adaptivity strategy is introduced,which is based on the high-order contribution calculated using hierarchical basis.Numerical results indicate that the hp-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper,the discontinuous Galerkin(DG)method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases.Firstly,the spectral property of the DG method is analyzed using the approximate dispersion relation(ADR)method and compared with typical finite difference methods,which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error.Then several typical test cases relevant to problems of compressible turbulence are simulated,including one-dimensional shock/entropy wave interaction,two-dimensional decaying isotropic turbulence,and two-dimensional temporal mixing layers.Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes.Furthermore,shocks are also well captured using the localized artificial diffusivity method.The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.

Numerical investigation of an active TPS for a wing leading edge exposed to high temperature air behind a strong bow shock wave

An array of subsonic counter-flow jets is studied as an active thermal protection system(TPS)for wing leading edges of hypersonic vehicles.The performance is numerically estimated in the model case of a circular cylinder on the basis of the 2D compressible Navier-Stokes equations.In contrast to a single subsonic jet,an array of jets is robust against variation of the angle of attack;high cooling effectiveness is confirmed up to 5°variation.The coolant gas(air)discharged from channels embedded in the cylinder covers over a wide range of the front surface of the cylinder.The feasibility of the active TPS is also discussed.

A 5th order monotonicity-preserving upwind compact difference scheme

Based on an upwind compact difference scheme and the idea of monotonicity-preserving, a 5th order monotonicity-preserving upwind compact difference scheme （m-UCD5） is proposed. The new difference scheme not only retains the advantage of good resolution of high wave number but also avoids the Gibbs phenomenon of the original upwind compact difference scheme. Compared with the classical 5th order WENO difference scheme, the new difference scheme is simpler and small in diffusion and computation load. By employing the component-wise and characteristic-wise methods, two forms of the new difference scheme are proposed to solve the N-S/Euler equation. Through the Sod problem, the Shu-Osher problem and tbe two-dimensional Double Mach Reflection problem, numerical solutions have demonstrated this new scheme does have a good resolution of high wave number and a robust ability of capturing shock waves, leading to a conclusion that the new difference scheme may be used to simulate complex flows containing shock waves.

A Priori Subcell Limiting Approach for the FR/CPR Method on Unstructured Meshes

A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction(FR/CPR)methods on twodimensional unstructured quadrilateralmeshes.Firstly,a modified indicator based on modal energy coefficients is proposed to detect troubled cells,where discontinuities exist.Then,troubled cells are decomposed into nonuniform subcells and each subcell has one solution point.A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted(NNW)interpolation is constructed to perform the calculation on troubled cells while smooth cells are calculated by the CPR method.Numerical investigations show that the proposed subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.

On Discreteness of the Hopf Equation

The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.

A Well-Balanced Weighted Compact Nonlinear Scheme for Pre-Balanced Shallow Water Equations

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.

AUSM-Based High-Order Solution for Euler Equations

In this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method(AUSM),specifically AUSM+-UP[9],with highorder upwind-biased interpolation procedures,theweighted essentially non-oscillatory(WENO-JS)scheme[8]and its variations[2,7],and the monotonicity preserving(MP)scheme[16],for solving the Euler equations.MP is found to be more effective than the three WENO variations studied.AUSM+-UP is also shown to be free of the so-called“carbuncle”phenomenon with the high-order interpolation.The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables,even though they require additional matrix-vector operations.Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison.In addition,four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy.Finally,a measure for quantifying the efficiency of obtaining high order solutions is proposed;the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

Synchronization of Weighted Essentially Non-Oscillatory Methods

Weighted essentially non-oscillatory(WENO)methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence.Under certain conditions in compressible turbulence,however,numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions.Of the nonlinear error that remains,we demonstrate that a large fraction is generated by a“synchronization deficiency”that interferes with the expression of theoretically predicted numerical performance characteristics when the WENO adaptation mechanism is engaged.This deficiency is illustrated numerically in simulations of a linearly advected sinusoidal wave and the Shu-Osher problem[J.Comput.Phys.,83(1989),pp.32-78].It is shown that attempting to correct this deficiency through forcible synchronization results in violation of conservation.We conclude that,for the given choice of candidate stencils,the synchronization deficiency cannot be adequately resolved under the current WENO smoothness measurement technique.