A conservative high order semi-Lagrangian WENO method for the Vlasov equation

In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially non-oscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear hyperbolic equations, can be factored into right and left flux matrices. It is the factoring of the interpolation matrices which makes it possible to apply the WENO methodology in the reconstruction used in the semi-Lagrangian update. The spatial WENO reconstruction developed for this method is conservative and updates point values of the solution. While the third, fifth, seventh and ninth order reconstructions are presented in this paper, the scheme can be extended to arbitrarily high order. WENO reconstruction is able to achieve high order accuracy in smooth parts of the solution while being able to capture sharp interfaces without introducing oscillations. Moreover, the CFL time step restriction of a regular finite difference or finite volume WENO scheme is removed in a semi-Lagrangian framework, allowing for a cheaper and more flexible numerical realization. The quality of the proposed method is demonstrated by applying the approach to basic test problems, such as linear advection and rigid body rotation, and to classical plasma problems, such as Landau damping and the two-stream instability. Even though the method is only second order accurate in time, our numerical results suggest the use of high order reconstruction is advantageous when considering the Vlasov–Poisson system.     

Artificial compressibility method revisited: Asymptotic numerical method for incompressible Navier–Stokes equations

The artificial compressibility method for the incompressible Navier–Stokes equations is revived as a high order accurate numerical method (fourth order in space and second order in time). Similar to the lattice Boltzmann method, the mesh spacing is linked to the Mach number. An accuracy higher than that of the lattice Boltzmann method is achieved by exploiting the asymptotic behavior of the solution of the artificial compressibility equations for small Mach numbers and the simple lattice structure. An easy method for accelerating the decay of acoustic waves, which deteriorate the quality of the numerical solution, and a simple cure for the checkerboard instability are proposed. The high performance of the scheme is demonstrated not only for the periodic boundary condition but also for the Dirichlet-type boundary condition.     

Shock-Capturing and Front-Tracking Methods for Granular Avalanches

Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shock-capturing numerical scheme for the one-dimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a front-tracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the front-tracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat run-out region, where a shock is formed as the avalanche comes to a halt.     

An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws

In this article we develop an improved version of the classical fifth-order weighted essentially non-oscillatory finite difference scheme of [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] (WENO-JS) for hyperbolic conservation laws. Through the novel use of a linear combination of the low order smoothness indicators already present in the framework of WENO-JS, a new smoothness indicator of higher order is devised and new non-oscillatory weights are built, providing a new WENO scheme (WENO-Z) with less dissipation and higher resolution than the classical WENO. This new scheme generates solutions that are sharp as the ones of the mapped WENO scheme (WENO-M) of Henrick et al. [A.K. Henrick, T.D. Aslam, J.M. Powers, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys. 207 (2005) 542–567], however with a 25% reduction in CPU costs, since no mapping is necessary. We also provide a detailed analysis of the convergence of the WENO-Z scheme at critical points of smooth solutions and show that the solution enhancements of WENO-Z and WENO-M at problems with shocks comes from their ability to assign substantially larger weights to discontinuous stencils than the WENO-JS scheme, not from their superior order of convergence at critical points. Numerical solutions of the linear advection of discontinuous functions and nonlinear hyperbolic conservation laws as the one dimensional Euler equations with Riemann initial value problems, the Mach 3 shock–density wave interaction and the blastwave problems are compared with the ones generated by the WENO-JS and WENO-M schemes. The good performance of the WENO-Z scheme is also demonstrated in the simulation of two dimensional problems as the shock–vortex interaction and a Mach 4.46 Richtmyer–Meshkov Instability (RMI) modeled via the two dimensional Euler equations.     

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.     

Implicit preconditioned WENO scheme for steady viscous flow computation

A class of lower–upper symmetric Gauss–Seidel implicit weighted essentially nonoscillatory (WENO) schemes is developed for solving the preconditioned Navier–Stokes equations of primitive variables with Spalart–Allmaras one-equation turbulence model. The numerical flux of the present preconditioned WENO schemes consists of a first-order part and high-order part. For first-order part, we adopt the preconditioned Roe scheme and for the high-order part, we employ preconditioned WENO methods. For comparison purpose, a preconditioned TVD scheme is also given and tested. A time-derivative preconditioning algorithm is devised and a discriminant is devised for adjusting the preconditioning parameters at low Mach numbers and turning off the preconditioning at intermediate or high Mach numbers. The computations are performed for the two-dimensional lid driven cavity flow, low subsonic viscous flow over S809 airfoil, three-dimensional low speed viscous flow over 6:1 prolate spheroid, transonic flow over ONERA-M6 wing and hypersonic flow over HB-2 model. The solutions of the present algorithms are in good agreement with the experimental data. The application of the preconditioned WENO schemes to viscous flows at all speeds not only enhances the accuracy and robustness of resolving shock and discontinuities for supersonic flows, but also improves the accuracy of low Mach number flow with complicated smooth solution structures.     

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.     

An Eulerian–Lagrangian WENO finite volume scheme for advection problems

We develop a locally conservative Eulerian–Lagrangian finite volume scheme with the weighted essentially non-oscillatory property (EL–WENO) in one-space dimension. This method has the advantages of both WENO and Eulerian–Lagrangian schemes. It is formally high-order accurate in space (we present the fifth order version) and essentially non-oscillatory. Moreover, it is free of a CFL time step stability restriction and has small time truncation error. The scheme requires a new integral-based WENO reconstruction to handle trace-back integration. A Strang splitting algorithm is presented for higher-dimensional problems, using both the new integral-based and pointwise-based WENO reconstructions. We show formally that it maintains the fifth order accuracy. It is also locally mass conservative. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy.     

多介质五方程简化模型及界面捕捉的人工压缩算法

根据两介质五方程简化模型的基本假设,发展了适用于任意多种介质的体积分数方程.为了捕捉多介质界面,将HLLC-HLLCM混合型数值通量的计算格式推广应用于二维平面和柱几何的多介质复杂流动问题,在高阶精度的数据重构过程中采用斜率修正型人工压缩方法ACM.通过一维、二维多介质黎曼问题算例测试,结果表明:发展的计算格式能够较好...     

Nonlinear spectral-like schemes for hybrid schemes

In spectral-like resolution-WENO hybrid schemes,if the switch function takes more grid points as discontinuity points,the WENO scheme is often turned on,and the numerical solutions may be too dissipative.Conversely,if the switch function takes less grid points as discontinuity points,the hybrid schemes usually are found to produce oscillatory solutions or just to be unstable.Even if the switch function takes less grid points as discontinuity points,the final hybrid scheme is inclined to be more stable,provided the spectral-like resolution scheme in the hybrid scheme has moderate shock-capturing capability.Following this idea,we propose nonlinear spectral-like schemes named weighted group velocity control(WGVC)schemes.These schemes show not only high-resolution for short waves but also moderate shock capturing capability.Then a new class of hybrid schemes is designed in which the WGVC scheme is used in smooth regions and the WENO scheme is used to capture discontinuities.These hybrid schemes show good resolution for small-scales structures and fine shock-capturing capabilities while the switch function takes less grid points as discontinuity points.The seven-order WGVC-WENO scheme has also been applied successfully to the direct numerical simulation of oblique shock wave-turbulent boundary layer interaction.     

A class of finite difference schemes with low dispersion and controllable dissipation for DNS of compressible turbulence

In this paper, a class of finite difference schemes which achieves low dispersion and controllable dissipation in smooth region and robust shock-capturing capabilities in the vicinity of discontinuities is presented. Firstly, a sufficient condition for semi-discrete finite difference schemes to have independent dispersion and dissipation is derived. This condition enables a novel approach to separately optimize the dissipation and dispersion properties of finite difference schemes and a class of schemes with minimized dispersion and controllable dissipation is thus obtained. Secondly, for the purpose of shock-capturing, one of these schemes is used as the linear part of the WENO scheme with symmetrical stencils to constructed an improved WENO scheme. At last, the improved WENO scheme is blended with its linear counterpart to form a new hybrid scheme for practical applications. The proposed scheme is accurate, flexible and robust. The accuracy and resolution of the proposed scheme are tested by the solutions of several benchmark test cases. The performance of this scheme is further demonstrated by its application in the direct numerical simulation of compressible turbulent channel flow between isothermal walls.     

An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces

This paper is devoted to developing a multi-material numerical scheme for non-linear elastic solids, with emphasis on the inclusion of interfacial boundary conditions. In particular for colliding solid objects it is desirable to allow large deformations and relative slide, whilst employing fixed grids and maintaining sharp interfaces. Existing schemes utilising interface tracking methods such as volume-of-fluid typically introduce erroneous transport of tangential momentum across material boundaries. Aside from combatting these difficulties one can also make improvements in a numerical scheme for multiple compressible solids by utilising governing models that facilitate application of high-order shock capturing methods developed for hydrodynamics. A numerical scheme that simultaneously allows for sliding boundaries and utilises such high-order shock capturing methods has not yet been demonstrated. A scheme is proposed here that directly addresses these challenges by extending a ghost cell method for gas-dynamics to solid mechanics, by using a first-order model for elastic materials in conservative form. Interface interactions are captured using the solution of a multi-material Riemann problem which is derived in detail. Several different boundary conditions are considered including solid/solid and solid/vacuum contact problems. Interfaces are tracked using level-set functions. The underlying single material numerical method includes a characteristic based Riemann solver and high-order WENO reconstruction. Numerical solutions of example multi-material problems are provided in comparison to exact solutions for the one-dimensional augmented system, and for a two-dimensional friction experiment.     

High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.     

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics

We present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio–Wu Riemann problems, the two-dimensional Kelvin–Helmholtz instability problems, and the two-dimensional Orszag–Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.     

基于WENO重构的熵稳定格式求解浅水方程

针对浅水方程,提出一种数值求解格式:空间方向采用满足熵稳定条件的数值通量,并在单元交界面处进行高阶WENO重构,时间上的推进采用强稳定的Runge-Kutta方法.模拟一维和二维经典问题,结果表明,该格式具有分辨率高、基本无振荡性等特点.     

On the characteristics-based ACM for incompressible flows

In this paper, the revised characteristics-based (CB) method for incompressible flows recently derived by Neofytou [P. Neofytou, Revision of the characteristic-based scheme for incompressible flows, J. Comput. Phys. 222 (2007) 475–484] has been further investigated. We have derived all the formulas for pressure and velocities from this revised CB method, which is based on the artificial compressibility method (ACM) [A.J. Chorin, A numerical solution for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967) 12]. Then we analyze the formulations of the original CB method [D. Drikakis, P.A. Govatsos, D.E. Papatonis, A characteristic based method for incompressible flows, Int. J. Numer. Meth. Fluids 19 (1994) 667–685; E. Shapiro, D. Drikakis, Non-conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows, Int. J. Numer. Meth. Eng. 66 (2006) 1466–1482; D. Drikakis, P.K. Smolarkiewicz, On spurious vortical structures, J. Comput. Phys. 172 (2001) 309–325; F. Mallinger, D. Drikakis, Instability in three-dimensional, unsteady stenotic flows, Int. J. Heat Fluid Flow 23 (2002) 657–663; E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Parts I. Derivation of different formulations and constant density limit, J. Comput. Phys. 210 (2005) 584–607; Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756] to investigate their consistency with the governing flow equations after convergence has been achieved. Furthermore we have implemented both formulations in an unstructured-grid finite volume solver [Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756]. Detailed numerical experiments show that both methods give almost identical solutions and convergence rates. Both can generate solutions which agree well with published results and experimental measurements. We thus conclude that both methods, being upwind schemes designed for the ACM, have the same performances in terms of accuracy and convergence speed, even though the revised method is more complex with less stringent assumptions made, while the original CB method is simpler due to the use of extra simplifying assumptions.     

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.     

气相爆轰高阶中心差分-WENO组合格式自适应网格方法

研究一种高阶中心差分-WENO组合格式,并采用自适应网格方法进行二维和三维气相爆轰波的数值模拟.采用ZND爆轰模型的控制方程为包含化学反应源项的Euler方程组.组合格式在大梯度区采用WENO格式捕捉间断,在光滑区采用高阶中心差分格式提高计算效率.采用一种基于流场结构特征的自适应网格.计算结果,表明这种方法同时具有高精度、高分辨率和高效率的特点.     

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework  and  with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge–Kutta (RK) method in time (WENO-RK)  and  by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.