The influence of cell geometry on the Godunov scheme applied to the linear wave equation

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier–Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.     

The influence of cell geometry on the accuracy of upwind schemes in the low mach number regime

It is well known, that standard upwind schemes for the Euler equations face a number of problems in the low Mach number regime: stiffness, cancellation and accuracy problems. A new aspect, presented in this paper, is the dependence on the cell geometry: applied on a triangular grid, the accuracy problem disappears, i.e. flows of arbitrarily small Mach numbers can be simulated on a fixed mesh. We give an asymptotic analysis of this, up to date unknown, phenomenon for the first-order Roe scheme and present a number of numerical results.     

A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability,implementation and tests

The ideal MHD equations are a central model in astrophysics, and their solution relies upon stable numerical schemes. We present an implementation of a new method, which possesses excellent stability properties. Numerical tests demonstrate that the theoretical stability properties are valid in practice with negligible compromises to accuracy. The result is a highly robust scheme with state-of-the-art efficiency. The scheme’s robustness is due to entropy stability, positivity and properly discretised Powell terms. The implementation takes the form of a modification of the MHD module in the FLASH code, an adaptive mesh refinement code. We compare the new scheme with the standard FLASH implementation for MHD. Results show comparable accuracy to standard FLASH with the Roe solver, but highly improved efficiency and stability, particularly for high Mach number flows and low plasma β. The tests include 1D shock tubes, 2D instabilities and highly supersonic, 3D turbulence. We consider turbulent flows with RMS sonic Mach numbers up to 10, typical of gas flows in the interstellar medium. We investigate both strong initial magnetic fields and magnetic field amplification by the turbulent dynamo from extremely high plasma β. The energy spectra show a reasonable decrease in dissipation with grid refinement, and at a resolution of 512³ grid cells we identify a narrow inertial range with the expected power law scaling. The turbulent dynamo exhibits exponential growth of magnetic pressure, with the growth rate higher from solenoidal forcing than from compressive forcing. Two versions of the new scheme are presented, using relaxation-based 3-wave and 5-wave approximate Riemann solvers, respectively. The 5-wave solver is more accurate in some cases, and its computational cost is close to the 3-wave solver.     

A discontinuous Galerkin method for inviscid low Mach number flows

In this work we extend the high-order discontinuous Galerkin (DG) finite element method to inviscid low Mach number flows. The method here presented is designed to improve the accuracy and efficiency of the solution at low Mach numbers using both explicit and implicit schemes for the temporal discretization of the compressible Euler equations. The algorithm is based on a classical preconditioning technique that in general entails modifying both the instationary term of the governing equations and the dissipative term of the numerical flux function (full preconditioning approach). In the paper we show that full preconditioning is beneficial for explicit time integration while the implicit scheme turns out to be efficient and accurate using just the modified numerical flux function. Thus the implicit scheme could also be used for time accurate computations. The performance of the method is demonstrated by solving an inviscid flow past a NACA0012 airfoil at different low Mach numbers using various degrees of polynomial approximations. Computations with and without preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers.     

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations

We compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: a first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.     

A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension

We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node based and compatible with the mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests problems show the efficiency of this approach. We also consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques at the end of this work. We detail the coefficients of the isoparametric element in the appendix.     

一维守恒的Lagrangian ADER格式

基于欧拉框架下ADER格式,构造一维守恒只有一个时间步的、高精度中心型拉格朗日ADER(LADER)格式.构造r阶LADER格式包括:从欧拉方程出发推导拉格朗日框架下积分形式的方程、采用WENO方法高精度重构节点处守恒量和从1阶到r-1阶的空间导数、求拉氏框架下这些变量的Godunov值,并计算1阶到r-1阶的时间全导数,最后高精度离散积分形式的流通量函数.对光滑流场的模拟表明,LADER格式达到设计的精度;对含强间断的流场模拟表明,数值解在间断附近基本无振荡.     

BGK方法在三维非结构网格上的初步应用

将BGK计算方法从二维拓展到三维并且应用于三维非结构网格,具有重要的理论价值和实用价值.采用旋转局部座标的方法,发展了一种针对三维非结构网格的BGK计算方法.在计算过程中,将最小二乘法应用于三维非结构网格的导数计算.对三维激波管和三维欠膨胀垂直冲击射流等两个算例进行了细致分析.这两个算例的计算结果表明,该方法在三维非结构网格上的初步应用是成功的 关键词： 气动BGK方法 三维 非结构网格     

A Free-Lagrange Augmented Godunov Method for the Simulation of Elastic–Plastic Solids

A Lagrangian finite-volume Godunov scheme is extended to simulate two-dimensional solids in planar geometry. The scheme employs an elastic–perfectly plastic material model, implemented using the method of radial return, and either the ‘stiffened’ gas or Osborne equation of state to describe the material. The problem of mesh entanglement, common to conventional two-dimensional Lagrangian schemes, is avoided by utilising the free-Lagrange Method. The Lagrangian formulation enables features convecting at the local velocity, such as material interfaces, to be resolved with minimal numerical dissipation. The governing equations are split into separate subproblems and solved sequentially in time using a time-operator split procedure. Local Riemann problems are solved using a two-shock approximate Riemann solver, and piecewise-linear data reconstruction is employed using a MUSCL-based approach to improve spatial accuracy. To illustrate the effectiveness of the technique, numerical simulations are presented and compared with results from commercial fixed-connectivity Lagrangian and smooth particle hydrodynamics solvers (AUTODYN-2D). The simulations comprise the low-velocity impact of an aluminium projectile on a semi-infinite target, the collapse of a thick-walled beryllium cylinder, and the high-velocity impact of cylindrical aluminium and steel projectiles on a thin aluminium target. The analytical solution for the collapse of a thick-walled cylinder is also presented for comparison.     

A low-Mach number fix for Roe’s approximate Riemann solver

We present a low-Mach number fix for Roe’s approximate Riemann solver (LMRoe). As the Mach number Ma tends to zero, solutions to the Euler equations converge to solutions of the incompressible equations. Yet, standard upwind schemes do not reproduce this convergence: the artificial viscosity grows like 1/Ma, leading to a loss of accuracy as Ma → 0. With a discrete asymptotic analysis of the Roe scheme we identify the responsible term: the jump in the normal velocity component ΔU of the Riemann problem. The remedy consists of reducing this term by one order of magnitude in terms of the Mach number. This is achieved by simply multiplying ΔU with the local Mach number. With an asymptotic analysis it is shown that all discrepancies between continuous and discrete asymptotics disappear, while, at the same time, checkerboard modes are suppressed. Low Mach number test cases show, first, that the accuracy of LMRoe is independent of the Mach number, second, that the solution converges to the incompressible limit for Ma → 0 on a fixed mesh and, finally, that the new scheme does not produce pressure checkerboard modes. High speed test cases demonstrate the fall back of the new scheme to the classical Roe scheme at moderate and high Mach numbers.     

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.     

Semi-Lagrangian Schemes for Hamilton–Jacobi Equations,Discrete Representation Formulae and Godunov Methods

We study a class of semi-Lagrangian schemes which can be interpreted as a discrete version of the Hopf–Lax–Oleinik representation formula for the exact viscosity solution of first order evolutive Hamilton–Jacobi equations. That interpretation shows that the scheme is potentially accurate to any prescribed order. We discuss how the method can be implemented for convex and coercive Hamiltonians with a particular structure and how this method can be coupled with a discrete Legendre trasform. We also show that in one dimension, the first-order semi-Lagrangian scheme coincides with the integration of the Godunov scheme for the corresponding conservation laws. Several test illustrate the main features of semi-Lagrangian schemes for evolutive Hamilton–Jacobi equations.     

Nondissipative and energy-stable high-order finite-difference interface schemes for 2-D patch-refined grids

A class of finite-difference interface schemes suitable for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution is presented. Grids of this type are generated by adaptive mesh refinement methods according to resolution needs dictated by the physics of the problem being modeled. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Three distinct geometries are identified at the interfaces of a domain with interior patch-refinement: edges, concave corners and convex corners. Asymptotic stability in time of the numerical scheme is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface stencils corresponding to an explicit fourth-order finite-difference scheme are presented for each geometry. To preserve stability, a reduction in local accuracy is required at the corner geometries. It is also found that no second-order accurate solution exists that satisfies the summation-by-parts condition. Tests using the 2-D scalar advection equation and an inviscid compressible vortex support the stability and accuracy of these stencils for both linear and nonlinear problems.     

A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a Multi-Implicit Spectral Deferred Correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework and allows us to conserve mass and energy while simultaneously satisfying the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.     

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.     

A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics

We present a highly robust second order accurate scheme for the Euler equations and the ideal MHD equations. The scheme is of predictor–corrector type, with a MUSCL scheme following as a special case. The crucial ingredients are an entropy stable approximate Riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure. For multidimensional MHD, a new discrete form of the Powell source terms is vital to ensure the stability properties. The numerical examples show that the scheme has superior stability compared to standard schemes, while maintaining accuracy. In particular, the method can handle very low values of pressure (i.e. low plasma β$\beta$ or high Mach numbers) and low mass densities.     

An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow

During the past decade gas-kinetic methods based on the BGK simplification of the Boltzmann equation have been employed to compute fluid flow in a finite-difference or finite-volume context. Among the most successful formulations is the finite-volume scheme proposed by Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (48) (2001) 289–335]. In this paper we build on this theoretical framework mainly with the aim to improve the efficiency and convergence of the scheme, and extend the range of application to three-dimensional complex geometries using general unstructured meshes. To that end we propose a modified BGK finite-volume scheme, which significantly reduces the computational cost, and improves the behavior on stretched unstructured meshes. Furthermore, a modified data reconstruction procedure is presented to remove the known problem that the Chapman–Enskog expansion of the BGK equation fixes the Prandtl number at unity. The new Prandtl number correction operates at the level of the partial differential equations and is also significantly cheaper for general formulations than previously published methods. We address the issue of convergence acceleration by applying multigrid techniques to the kinetic discretization. The proposed modifications and convergence acceleration help make large-scale computations feasible at a cost competitive with conventional discretization techniques, while still exploiting the advantages of the gas-kinetic discretization, such as computing full viscous fluxes for finite volume schemes on a simple two-point stencil.     

Block structured adaptive mesh and time refinement for hybrid,hyperbolic + N-body systems

We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov’s method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.     

Quasisteady aero-acoustic response of orifices.

The low frequency response of orifices (slit, circular diaphragm, and perforated plate) in the presence of mean flow is well predicted by a quasisteady theory. A refinement is brought to the theory by considering a Mach number dependent vena contracta coefficient. The measurements of the vena contracta coefficient of a slit agree well with the simple analytical expression existing in the case of the Borda tube orifice. The measured scattering matrix coefficients do not depend strongly on the geometry of the element. If the frequency is increased the moduli remain relatively unaffected while the arguments exhibit a complex behavior which depends on the geometry. From these considerations an anechoic termination efficient at high mass flow is designed.     

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.