Upwind schemes for the wave equation in second-order form

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.     

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.     

Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

The goal of this paper is to generalize the well-balanced approach for non-equilibrium flow studied by Wang et al. (2009) [29] to a class of low dissipative high-order shock-capturing filter schemes and to explore more advantages of well-balanced schemes in reacting flows. More general 1D and 2D reacting flow models and new examples of shock turbulence interactions are provided to demonstrate the advantage of well-balanced schemes. The class of filter schemes developed by Yee et al. (1999) [33], Sjögreen and Yee (2004) [27] and Yee and Sjögreen (2007) [38] consist of two steps, a full time step of spatially high-order non-dissipative base scheme and an adaptive non-linear filter containing shock-capturing dissipation. A good property of the filter scheme is that the base scheme and the filter are stand-alone modules in designing. Therefore, the idea of designing a well-balanced filter scheme is straightforward, i.e. choosing a well-balanced base scheme with a well-balanced filter (both with high-order accuracy). A typical class of these schemes shown in this paper is the high-order central difference schemes/predictor–corrector (PC) schemes with a high-order well-balanced WENO filter. The new filter scheme with the well-balanced property will gather the features of both filter methods and well-balanced properties: it can preserve certain steady-state solutions exactly; it is able to capture small perturbations, e.g. turbulence fluctuations; and it adaptively controls numerical dissipation. Thus it shows high accuracy, efficiency and stability in shock/turbulence interactions. Numerical examples containing 1D and 2D smooth problems, 1D stationary contact discontinuity problem and 1D turbulence/shock interactions are included to verify the improved accuracy, in addition to the well-balanced behavior.     

Analysis of numerical dispersion properties of SFDTD and MRTD schemes

In this paper, the Maxwell's equations are written as Hamilton canonical equations by using Hamilton functional variation method. Maxwell's equations can be discretized with symplectic propagation technique combined with high-order difference schemes approximations to construct symplectic finite difference time domain (SFDTD) method. The high-order dispersion equations of the scheme for space is deduced. The numerical dispersion analysis is included, and it is compared with the multiresolution time-domain (MRTD) method based on the Daubechies scaling functions. Numerical results show high efficiency and accuracy of the SFDTD method.     

Solvers for the high-order Riemann problem for hyperbolic balance laws

We study three methods for solving the Cauchy problem for a system of non-linear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin, a high-order Riemann problem. Two of the methods are new; one of the them results from a re-interpretation of the high-order numerical methods proposed by Harten et al. [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231–303] and the other is a modification of the solver in [E.F. Toro, V.A. Titarev, Solution of the generalised Riemann problem for advection-reaction equations, Proc. Roy. Soc. London A 458 (2002) 271–281]. A systematic assessment of all three solvers is carried out and their relative merits are discussed. We also implement the solvers, locally, in the context of high-order finite volume numerical methods of the ADER type, on unstructured meshes. Schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations and the shallow water equations with source terms are constructed. Empirically obtained convergence rates are studied systematically and, for the tests considered, these correspond to the theoretically expected orders of accuracy. We also address the question of balance between flux gradients and source terms, for steady flow. We find that the ADER schemes may be termed asymptotically well-balanced, in the sense that the well-balanced property is attained as the order of the method increases, and this without introducing any ad-hoc fixes to the schemes or the equations.     

Compact upwind schemes on adaptive octrees

Compact high-order upwind schemes using reconstruction from cell-averages are derived for application with the compressible three-dimensional Navier–Stokes equations. An adaptive-octree mesh, combined with the Adams–Bashforth–Moulton family of predictor–corrector schemes, provides a conservative high-order time-integration platform supporting localized h-refinement and timestep sub-cycling. Numerical examples for smooth flows demonstrate the improvement over explicit upwind schemes and formal accuracy of the schemes, as well as the behavior in wall-bounded regions, and the resolution of a broad wavenumber spectrum.     

Compact finite volume schemes on boundary-fitted grids

The paper focuses on the development of a framework for high-order compact finite volume discretization of the three-dimensional scalar advection–diffusion equation. In order to deal with irregular domains, a coordinate transformation is applied between a curvilinear, non-orthogonal grid in the physical space and the computational space. Advective fluxes are computed by the fifth-order upwind scheme introduced by Pirozzoli [S. Pirozzoli, Conservative hybrid compact-WENO schemes for shock–turbulence interaction, J. Comp. Phys. 178 (2002) 81] while the Coupled Derivative scheme [M.H. Kobayashi, On a class of Padé finite volume methods, J. Comp. Phys. 156 (1999) 137] is used for the discretization of the diffusive fluxes.Numerical tests include unsteady diffusion over a distorted grid, linear free-surface gravity waves in a irregular domain and the advection of a scalar field. The proposed methodology attains high-order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales.     

A stable interface element scheme for the p-adaptive lifting collocation penalty formulation

This paper presents a procedure for adaptive polynomial refinement in the context of the lifting collocation penalty (LCP) formulation. The LCP scheme is a high-order unstructured discretization method unifying the discontinuous Galerkin, spectral volume, and spectral difference schemes in single differential formulation. Due to the differential nature of the scheme, the treatment of inter-cell fluxes for spatially varying polynomial approximations is not straightforward. Specially designed elements are proposed to tackle non-conforming polynomial approximations. These elements are constructed such that a conforming interface between polynomial approximations of different degrees is recovered. The stability and conservation properties of the scheme are analyzed and various inviscid compressible flow calculations are performed to demonstrate the potential of the proposed approach.     

Artificial boundary conditions for the numerical solution of the Euler equations by the discontinuous galerkin method

We present artificial boundary conditions for the numerical simulation of compressible flows using high-order accurate discretizations with the discontinuous Galerkin (DG) finite element method. The construction of the proposed boundary conditions is based on characteristic analysis and applied for boundaries with arbitrary shape and orientation. Numerical experiments demonstrate that the proposed boundary treatment enables to convect out of the computational domain complex flow features with little distortion. In addition, it is shown that small-amplitude acoustic disturbances could be convected out of the computational domain, with no significant deterioration of the overall accuracy of the method. Furthermore, it was found that application of the proposed boundary treatment for viscous flow over a cylinder yields superior performance compared to simple extrapolation methods.     

迎风紧致格式求解Hamilton-Jacobi方程

基于Hamilton-Jacobi(H-J)方程和双曲型守恒律之间的关系,将三阶和五阶迎风紧致格式推广应用于求解H-J方程,建立了高精度的H-J方程求解方法.给出了一维和二维典型数值算例的计算结果,其中包括一个平面激波作用下的Richtmyer Meshkov界面不稳定性问题.数值试验表明,在解的光滑区域该方法具有高精度,而在导数不连续的不光滑区域也获得了比较好的分辨效果.相比于同阶精度的WENO格式,本方法具有更小的数值耗散,从而有利于多尺度复杂流动的模拟中H-J方程的求解.     

一种混合优化差分格式

优化差分格式一般用于计算气动声学和小尺度的湍流数值模拟,这类格式为了获取更好的短波分辨率通常牺牲了部分收敛精度.文章尝试结合最高阶精度格式与优化格式的特点,构造混合优化格式,提高优化格式的收敛精度以及谱分辨率.混合优化格式由模板上的最高阶精度格式与优化格式加权组合得到,权系数由当前模板上的值确定,这使得该格式为非线性格式.对于单色波问题,通过优化权的设计可大幅度减小相位误差.但是加权混合过程使得计算时间有所增加.数值计算证明了该格式的特点.      

On accuracy and performance of high-order finite volume methods in local mean energy model of non-thermal plasmas

In this paper, a high-order finite volume method is employed to solve the local energy approximation model equations for a radio-frequency plasma discharge in a one-dimensional geometry. The so called deferred correction technique, along with high-order Lagrange polynomials, is used to calculate the convection and diffusion fluxes. Temporal discretization is performed using backward difference schemes of first and second orders. Extensive numerical experiments are carried out to evaluate the order and level of accuracy as well as computational efficiency of the various methods implemented in the work. These tests exhibit global convergence rate of up to fourth order for the spatial error, and of up to second order for the temporal error.     

Stabilized non-dissipative approximations of Euler equations in generalized curvilinear coordinates

We discuss stabilization strategies for finite-difference approximations of the compressible Euler equations in generalized curvilinear coordinates that do not rely on explicit upwinding or filtering of the physical variables. Our approach rather relies on a skew-symmetric-like splitting of the convective derivatives, that guarantees preservation of kinetic energy in the semi-discrete, low-Mach-number limit. A locally conservative formulation allows efficient implementation and easy incorporation into existing compressible flow solvers. The validity of the approach is tested for benchmark flow cases, including the propagation of a cylindrical vortex, and the head-on collision of two vortex dipoles. The tests support high accuracy and superior stability over conventional central discretization of the convective derivatives. The potential use for DNS/LES of turbulent compressible flows in complex geometries is discussed.     

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319–338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC–WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.     

基于WENO重构的高阶移动网格动理学格式

提出一种基于WENO重构的高阶(至少三阶)移动网格动理学格式.利用流体力学方程的积分形式得到移动网格上离散格式,再利用自适应移动网格方法移动网格,进而得到网格速度,利用WENO重构得到高阶插值多项式,最后使用时间方向上精确的动理学数值方法构造数值通量,得到移动网格单元上新的物理量.数值实验表明这种格式同时具有高精度、高分辨率的特点.     

WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions

The paper extends weighted essentially non-oscillatory (WENO) methods to three dimensional mixed-element unstructured meshes, comprising tetrahedral, hexahedral, prismatic and pyramidal elements. Numerical results illustrate the convergence rates and non-oscillatory properties of the schemes for various smooth and discontinuous solutions test cases and the compressible Euler equations on various types of grids. Schemes of up to fifth order of spatial accuracy are considered.     

Numerical methods for solid mechanics on overlapping grids: Linear elasticity

This paper presents a new computational framework for the simulation of solid mechanics on general overlapping grids with adaptive mesh refinement (AMR). The approach, described here for time-dependent linear elasticity in two and three space dimensions, is motivated by considerations of accuracy, efficiency and flexibility. We consider two approaches for the numerical solution of the equations of linear elasticity on overlapping grids. In the first approach we solve the governing equations numerically as a second-order system (SOS) using a conservative finite-difference approximation. The second approach considers the equations written as a first-order system (FOS) and approximates them using a second-order characteristic-based (Godunov) finite-volume method. A principal aim of the paper is to present the first careful assessment of the accuracy and stability of these two representative schemes for the equations of linear elasticity on overlapping grids. This is done by first performing a stability analysis of analogous schemes for the first-order and second-order scalar wave equations on an overlapping grid. The analysis shows that non-dissipative approximations can have unstable modes with growth rates proportional to the inverse of the mesh spacing. This new result, which is relevant for the numerical solution of any type of wave propagation problem on overlapping grids, dictates the form of dissipation that is needed to stabilize the scheme. Numerical experiments show that the addition of the indicated form of dissipation and/or a separate filter step can be used to stabilize the SOS scheme. They also demonstrate that the upwinding inherent in the Godunov scheme, which provides dissipation of the appropriate form, stabilizes the FOS scheme. We then verify and compare the accuracy of the two schemes using the method of analytic solutions and using problems with known solutions. These latter problems provide useful benchmark solutions for time dependent elasticity. We also consider two problems in which exact solutions are not available, and use a posterior error estimates to assess the accuracy of the schemes. One of these two problems is additionally employed to demonstrate the use of dynamic AMR and its effectiveness for resolving elastic “shock” waves. Finally, results are presented that compare the computational performance of the two schemes. These demonstrate the speed and memory efficiency achieved by the use of structured overlapping grids and optimizations for Cartesian grids.     

Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms

In [16], [17], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions), it is more difficult to design high order schemes which do not produce negative density or pressure. In this paper, we first show that our framework to construct positivity-preserving high order schemes in [16], [17] can also be applied to Euler equations with a general equation of state. Then we discuss an extension to Euler equations with source terms. Numerical tests of the third order Runge–Kutta DG (RKDG) method for Euler equations with different types of source terms are reported.     

Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier–Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourth-order differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.     

High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions

We present a space–time adaptive solver for single- and multi-phase compressible flows that couples average interpolating wavelets with high-order finite volume schemes. The solver introduces the concept of wavelet blocks, handles large jumps in resolution and employs local time-stepping for efficient time integration. We demonstrate that the inherently sequential wavelet-based adaptivity can be implemented efficiently in multicore computer architectures using task-based parallelism and introducing the concept of wavelet blocks. We validate our computational method on a number of benchmark problems and we present simulations of shock-bubble interaction at different Mach numbers, demonstrating the accuracy and computational performance of the method.