A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

High order finite difference methods with subcell resolution for advection equations with stiff source terms

A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.     

Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., [6], [10]. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in [1], [2] can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in [22]. In this paper, we first discuss an extension of the technique in [22], [23], [24] to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in [22]. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.     

Positivity-preserving high order finite difference WENO schemes for compressible Euler equations

In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.     

A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS

In this paper, a reactive dynamic user equilibrium model is extended to simulate two groups of pedestrians traveling on crossing paths in a continuous walking facility. Each group makes path choices to minimize the travel cost to its destination in a reactive manner based on instantaneous information. The model consists of a conservation law equation coupled with an Eikonal-type equation for each group. The velocity-density relationship of pedestrian movement is obtained via an experimental method. The model is solved using a finite volume method for the conservation law equation and a fast-marching method for the Eikonal-type equation on unstructured grids. The numerical results verify the rationality of the model and the validity of the numerical method. Based on this continuum model, a number of results, e.g., the formation of strips or moving clusters composed of pedestrians walking to the same destination, are also observed.     

ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES Ⅱ:CELL AVERAGES AND MULTIRESOLUTION

This paper is the second part of the article and is devoted to the construction and analysis of new non-linear optimal weights for WENO interpolation capable of rising the order of accuracy close to discontinuities for data discretized in the cell averages. Thus, now we are interested in analyzing the capabilities of the new algorithm when working with functions belonging to the subspace $L^1\cap L^2$ and that, consequently, are piecewise smooth and can present jump discontinuities. The new non-linear optimal weights are redesigned in a way that leads to optimal theoretical accuracy close to the discontinuities and at smooth zones. We will present the new algorithm for the approximation case and we will analyze its accuracy. Then we will explain how to use the new algorithm in multiresolution applications for univariate and bivariate functions. The numerical results confirm the theoretical proofs presented.     

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.     

On positivity preserving finite volume schemes for Euler equations

Summary. We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described. Received May 20, 1994     

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.     

High-order well-balanced schemes and applications to non-equilibrium flow

The appearance of the source terms in modeling non-equilibrium flow problems containing finite-rate chemistry or combustion poses additional numerical difficulties beyond that for solving non-reacting flows. A well-balanced scheme, which can preserve certain non-trivial steady state solutions exactly, may help minimize some of these difficulties. In this paper, a simple one-dimensional non-equilibrium model with one temperature is considered. We first describe a general strategy to design high-order well-balanced finite-difference schemes and then study the well-balanced properties of the high-order finite-difference weighted essentially non-oscillatory (WENO) scheme, modified balanced WENO schemes and various total variation diminishing (TVD) schemes. The advantages of using a well-balanced scheme in preserving steady states and in resolving small perturbations of such states will be shown. Numerical examples containing both smooth and discontinuous solutions are included to verify the improved accuracy, in addition to the well-balanced behavior.