High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

To improve the spectral characteristics of the high-order weighted compact nonlinear scheme(WCNS),optimized flux difference schemes are proposed.The disadvantages in previous optimization routines,i.e.,reducing formal orders,or extending stencil widths,are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes.Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation(ADR)for nonlinear schemes.Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators.The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes,in terms of accuracy and resolution for smooth wave and vortex,especially for long-time simulations.Using optimized schemes increases the total CPU time by less than 4%.

Improvement of Convergence to Steady State Solutions of Euler Equations with Weighted Compact Nonlinear Schemes

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes （WCNS） [Deng X. and Zhang H. （2000）, J. Comput. Phys. 165, 22–44 and Zhang S., Jiang S. and Shu C.-W. （2008）, J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. （2007）, J. Sci. Comput. 31, 273–305 and Zhang S., Jiang S and Shu. C.-W. （2011）, J. Sci. Comput. 47, 216–238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

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.

Assessment of Two Turbulence Models and Some Compressibility Corrections for Hypersonic Compression Corners by High-order Difference Schemes

The Spalart-Allmaras （S-A） turbulence model, the shear-stress transport （SST） turbulence model and their compressibility corrections are revaluated for hypersonic compression comer flows by using high-order difference schemes. The compressibility effect of density gradient, pressure dilatation and turbulent Mach number is accounted. In order to reduce confusions between model uncertainties and discretization errors, the formally fifth-order explicit weighted compact nonlinear scheme （WCNS-E-5） is adopted for convection terms, and a fourth-order staggered central difference scheme is applied for viscous terms. The 15° and 34° compression comers at Mach number 9.22 are investigated. Numerical results show that the original SST model is superior to the original S-A model in the resolution of separated regions and predictions of wall pressures and wall heat-flux rates. The capability of the S-A model can be largely improved by blending Catris＇ and Shur＇s compressibility corrections. Among the three corrections of the SST model listed in the present paper, Catris＇ modification brings the best results. However, the dissipation and pressure dilatation corrections result in much larger separated regions than that of the experiment, and are much worse than the original SST model as well as the other two corrections. The correction of turbulent Mach number makes the separated region slightly smaller than that of the original SST model. Some results of low-order schemes are also presented. When compared to the results of the high-order schemes, the separated regions are smaller, and the peak wall pressures and peak heat-flux rates are lower in the region of the reattachment points.

Assessment of alternative scale-providing variables in a Reynolds-stress model using high-order methods

This paper is set in the high-order finite-difference discretization of the Reynolds-averaged Navier-Stokes(RANS)equations,which are coupled with the turbulence model equations.Three alternative scale-providing variables for the specific dissipation rate(o)are implemented in the framework of the Reynolds stress model(RSM)for improving its robustness.Specifically,g(=1/√ω)has natural boundary conditions and reduced spatial gradients,and a new numerical constraint is imposed on itω(=lnω)can preserve positivity and also has reduced spatial gradients;the eddy viscosity v,also has natural boundary conditions and its equation is improved in this work.The solution polynomials of the mean-flow and turbulence-model equations are both reconstructed by the weighted compact nonlinear scheme(WCNS).Moreover,several numerical techniques are introduced to improve the numerical stability of the equation system.A range of canonical as well as industrial turbulent flows are simulated to assess the accuracy and robustness of the scale-transformed models.Numerical results show that the scale-transformed models have significantly improved robustness compared to the w model and still keep the characteristics of RSM.Therefore,the high-order discretization of the RANS and RSM equations,which number 12 in total,can be successfully achieved.