Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for?Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.     

Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method,III: Unstructured Meshes

In [J. Comput. Phys. 193:115–135, 2004] and [Comput. Fluids 34:642–663, 2005], Qiu and Shu developed a class of high order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems, and applied them as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods on structured meshes. In this continuation paper, we extend the method to solve two dimensional problems on unstructured meshes. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods. The research was partially supported by the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations, NSFC grant 10671091 and JSNSF BK2006511.     

Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes

The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting (Jiang and Shu, in J. Comput. Phys. 126:202–228, 1996) is investigated. Numerical evidence in Zhang and Shu (J. Sci. Comput. 31:273–305, 2007) indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the “essentially non-oscillatory” property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Differently from the strategy adopted in Zhang and Shu (J. Sci. Comput. 31:273–305, 2007), in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post-shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post-shock oscillation appears. If we instead use upwind-biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post-shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.     

Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation

In this paper, we present hybrid weighted essentially non-oscillatory (WENO) schemes with several discontinuity detectors for solving the compressible ideal magnetohydrodynamics (MHD) equation. Li and Qiu (J Comput Phys 229:8105–8129, 2010) examined effectiveness and efficiency of several different troubled-cell indicators in hybrid WENO methods for Euler gasdynamics. Later, Li et al. (J Sci Comput 51:527–559, 2012) extended the hybrid methods for solving the shallow water equations with four better indicators. Hybrid WENO schemes reduce the computational costs, maintain non-oscillatory properties and keep sharp transitions for problems. The numerical results of hybrid WENO-JS/WENO-M schemes are presented to compare the ability of several troubled-cell indicators with a variety of test problems. The focus of this paper, we propose optimal and reliable indicators for performance comparison of hybrid method using troubled-cell indicators for efficient numerical method of ideal MHD equations. We propose a modified ATV indicator that uses a second derivative. It is advantageous for differential discontinuity detection such as jump discontinuity and kink. A detailed numerical study of one-dimensional and two-dimensional cases is conducted to address efficiency (CPU time reduction and more accurate numerical solution) and non-oscillatory property problems. We demonstrate that the hybrid WENO-M scheme preserves the advantages of WENO-M and the ratio of computational costs of hybrid WENO-M and hybrid WENO-JS is smaller than that of WENO-M and WENO-JS.     

A sixth-order finite difference WENO scheme for Hamilton–Jacobi equations

In this paper, a sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed to approximate the viscosity solution of the Hamilton–Jacobi equations. This new WENO scheme has the same spatial nodes as the classical fifth-order WENO scheme proposed by Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM. J. Sci. Comput. 21 (2000), pp. 2126–2143] but can be as high as sixth-order accurate in smooth region while keeping sharp discontinuous transitions with no spurious oscillations near discontinuities. Extensive numerical experiments in one- and two-dimensional cases are carried out to illustrate the capability of the proposed scheme.     

Computations of steady and unsteady transport of pollutant in shallow water

We present new models for simulating the steady and unsteady transport of pollutant. Then the simple central-upwind schemes based on central weighted essentially non-oscillatory reconstructions are proposed in this paper for computing the one- and two-dimensional steady and unsteady models. Since the non-uniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Synchronously, due to the central-upwind schemes are combined with the fourth-order central weighted essentially non-oscillatory reconstructions, the schemes have the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our methods.     

Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.     

High-order mixed weighted compact and non-compact scheme for shock and small length scale interaction

It is critical for a numerical scheme to obtain numerical results as accurate as possible with limited computational resources. Turbulent processes are very sensitive to numerical dissipation, which may dissipate the small length scales. On the other hand, dealing with shock waves, capturing and reproducing of the discontinuity may lead to non-physical oscillations for non-dissipative high-order schemes. In the present work, a new high-order mixed weighted compact and non-compact difference scheme (MWCS hereafter) is proposed for accurate approximation of the derivatives in the governing Euler equations. The basic idea is to recover the non-dissipative high-order weighted compact scheme (WCS) in smooth regions, while linearly combine the WCS with a non-compact scheme, the weighted essentially non-oscillatory (WENO) scheme, for near-shock areas, by using a shock-detecting function. The proposed formulation does not involve any case-dependent adjustable parameter. A detailed Fourier and local truncation error analysis are used for assessing the dispersion and dissipation characteristics of the scheme. Numerical tests are performed for the one- and two-dimensional case and the results are compared with the well-established WENO scheme and the WCS.     

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.     

Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields

Euler equations of compressible gas dynamics, coupled with a source term due to the gravitational fields, often appear in many interesting astrophysical and atmospheric applications. In this paper, we design high order finite difference weighted essentially non-oscillatory (WENO) methods for the Euler equations under static gravitation fields, which are well-balanced for known steady state solutions. We simplify the well-balanced WENO methods designed in Xing and Shu (2013) for the isothermal equilibrium, and then extend them to more general steady state solutions which include both isothermal and polytropic equilibria. One- and two-dimensional numerical examples are provided at the end to test the performance of the proposed WENO methods and verify these properties numerically.     

A modified fifth-order WENO scheme for hyperbolic conservation laws

Recently a WENO scheme, with smoothness indicators constructed based on $L_1$ measure is introduced by Ha et al. (2013) and the improved version of this scheme is presented by Kim et al. (2016), referred to as WENO-NS and WENO-P schemes respectively. These schemes perform better than the existing many fifth-order WENO schemes for the problems which contain discontinuities and attain fifth-order accuracy at the critical points where the first derivative vanishes but not at the points where the second derivatives are zero. This paper deals with modification of the above said methods to obtain a new fifth-order weighted essentially non-oscillatory (WENO) scheme. A new global-smoothness indicator is proposed which shows an improved behavior over the solutions of WENO-NS and WENO-P schemes and the proposed scheme attains an optimal order of approximation, even at the critical points where the first and second derivatives vanish but not the third derivative. Examples are taken in the numeric section to check the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.     

A New Smoothness Indicator for the WENO Schemes and Its Effect on the Convergence to Steady State Solutions

The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax–Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202–228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the “essentially non-oscillatory” property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202–228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries. Dedicated to the memory of Professor Xu-Dong Liu.     

Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes

In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.     

A New Mapped Weighted Essentially Non-oscillatory Scheme

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Recently Henrick et al. (J. Comput. Phys. 207:542–567, 2005) noted that the fifth-order WENO method by Jiang and Shu (J. Comput. Phys. 126:202–228, 1996) is only third-order accurate near critical points of the smooth regions in general. Using a simple mapping function to the original weights in Jiang and Shu (J. Comput. Phys. 126:202–228, 1996), Henrick et al. developed a mapped WENO method to achieve the optimal order of accuracy near critical points. In this paper we study the mapped WENO scheme and find that, when it is used for solving the problems with discontinuities, the mapping function in Henrick et al. (J. Comput. Phys. 207:542–567, 2005) may amplify the effect from the non-smooth stencils and thus cause a potential loss of accuracy near discontinuities. This effect may be difficult to be observed for the fifth-order WENO method unless a long time simulation is desired. However, if the mapping function is applied to seventh-order WENO methods (Balsara and Shu in J. Comput. Phys. 160:405–452, 2000), the error can increase much faster so that it can be observed with a moderate output time. In this paper a new mapping function is proposed to overcome this potential loss of accuracy.     

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge–Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.     

解Hamilton-Jacobi方程的无振荡局部加密方法

０．引 言 近年来,Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程（简称Ｈ－Ｊ方程）的数学理论与数值逼近已引起人们越来越多的关注．Ｈ－Ｊ方程不仅在原有的领域例如控制论、微分几何等有非常重要的应用［８］,而且不断开拓新的应用领域,例如用于网格生成［５］以及流体界面的水平集方法计算 ［９,１２,１３,１５］等．由于 Ｈ－Ｊ方程解的导数会出现间断,导致解曲面（线）出现尖点或纽结等现象［７］,故如何做到既节省计算时间,又能在光滑区域高精度数值求解和较好地分辨间断是一个十分重要的问题．文卜］通过在每个坐标方向构造单变量的…     

Finite Volume HWENO Schemes for Nonconvex Conservation Laws

Following the previous work of Qiu and Shu (SIAM J Sci Comput 31: 584–607, 2008), we investigate the performance of Hermite weighted essentially non-oscillatory (HWENO) scheme for nonconvex conservation laws. Similar to many other high order methods, we show that the finite volume HWENO scheme performs poorly for some nonconvex conservation laws. We modify the scheme around the nonconvex regions, based on a first order monotone scheme and a second entropic projection, to ensure entropic convergence. Extensive numerical tests are performed. Compare with the earlier work of Qiu and Shu which focuses on 1D scalar problems, we apply the modified schemes (both WENO and HWENO) to one-dimensional Euler system with nonconvex equation of state and two-dimensional problems.     

On the Long-Time <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>-Stability of the Implicit Euler Scheme for the 2D Magnetohydrodynamics Equations

Pursuing our work in Tone (Asymptot. Analysis 51:231–245, 2007) and Tone and Wirosoetisno (SIAM J. Number. Analysis 44:29–40, 2006), we consider in this article the two-dimensional magnetohydrodynamics equations, we discretize these equations in time using the implicit Euler scheme and with the aid of the classical and uniform discrete Gronwall lemma, we prove that the scheme is H ²-uniformly stable in time.