A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. The scheme is designed to be able to preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area-weighted schemes that possess the spherical symmetry property, our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density, momentum and total energy. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry, accuracy and non-oscillatory properties.     

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.     

An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model

We propose a discretization method of a five-equation model with isobaric closure for the simulation of interfaces between compressible fluids. This numerical solver is a Lagrange–Remap scheme that aims at controlling the numerical diffusion of the interface between both fluids. This method does not involve any interface reconstruction procedure. The solver is equipped with built-in stability and consistency properties and is conservative with respect to mass, momentum, total energy and partial masses. This numerical scheme works with a very broad range of equations of state, including tabulated laws. Properties that ensure a good treatment of the Riemann invariants across the interface are proven. As a consequence, the numerical method does not create spurious pressure oscillations at the interface. We show one-dimensional and two-dimensional classic numerical tests. The results are compared with the approximate solutions obtained with the classic upwind Lagrange–Remap approach, and with experimental and previously published results of a reference test case.     

High-order finite-volume methods for the shallow-water equations on the sphere

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM⁺-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge–Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM⁺-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.     

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.     

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

The space–time CESE method for solving special relativistic hydrodynamic equations

The special relativistic hydrodynamic equations are more complicated than the classical ones due to the nonlinear and implicit relations that exist between conservative and primitive variables. In this article, a space–time conservation element and solution element (CESE) method is proposed for solving these equations in one and two space dimensions. The CESE method has capability to capture sharp propagating wavefront of the relativistic fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. The scheme is efficient, robust, and gives results comparable to those obtained with more sophisticated algorithms, even in highly relativistic two-dimensional test problems.     

Multidimensional HLLE Riemann solver: Application to Euler and magnetohydrodynamic flows

In this work we present a general strategy for constructing multidimensional HLLE Riemann solvers, with particular attention paid to detailing the two-dimensional HLLE Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow.We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical (MHD) flows. A robust and efficient second order accurate numerical scheme for two and three-dimensional Euler and MHD flows is presented. The scheme is built on the current multidimensional Riemann solver and has been implemented in the author’s RIEMANN code. The number of zones updated per second by this scheme on a modern processor is shown to be cost-competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps.Accuracy analysis for multidimensional Euler and MHD problems shows that the scheme meets its design accuracy. Several stringent test problems involving Euler and MHD flows are also presented and the scheme is shown to perform robustly on all of them.     

Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics

This work presents a multi-dimensional cell-centered unstructured finite volume scheme for the solution of multimaterial compressible fluid flows written in the Lagrangian formalism. This formulation is considered in the Arbitrary-Lagrangian–Eulerian (ALE) framework with the constraint that the mesh velocity and the fluid velocity coincide. The link between the vertex velocity and the fluid motion is obtained by a formulation of the momentum conservation on a class of multi-scale encased volumes around mesh vertices. The vertex velocity is derived with a nodal Riemann solver constructed in such a way that the mesh motion and the face fluxes are compatible. Finally, the resulting scheme conserves both momentum and total energy and, it satisfies a semi-discrete entropy inequality. The numerical results obtained for some classical 2D and 3D hydrodynamic test cases show the robustness and the accuracy of the proposed algorithm.     

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.     

一维多介质可压缩流动数值方法

应用高精度界面追踪方法计算一般状态方程的多介质可压缩流动问题;应用LevelSet技术捕捉界面位置,在界面附近采用守恒数值离散,用双波近似求解一般状态方程Riemann问题,并采用统一高阶PPM格式进行内点和交界面点的计算.一维算例表明,该方法对于光滑区域以及多介质交界面具有二阶精度,能准确地模拟交界面的位置,交界面计算无数值振荡和数值耗散,并能处理一般状态方程的多介质可压缩流动问题.     

An interface treating technique for compressible multi-medium flow with Runge–Kutta discontinuous Galerkin method

The high-order accurate Runge–Kutta discontinuous Galerkin (RKDG) method is applied to the simulation of compressible multi-medium flow, generalizing the interface treating method given in Chertock et al. (2008) [9]. In mixed cells, where the interface is located, Riemann problems are solved to define the states on both sides of the interface. The input states to the Riemann problem are obtained by extrapolation to the cell boundary from solution polynomials in the neighbors of the mixed cell. The level set equation is solved by using a high-order accurate RKDG method for Hamilton–Jacobi equations, resulting in a unified DG solver for the coupled problem. The method is conservative if we include the states in the mixed cells, which are however not used in the updating of the numerical solution in other cells. The states in the mixed cells are plotted to better evaluate the conservation errors, manifested by overshoots/undershoots when compared with states in neighboring cells. These overshoots/undershoots in mixed cells are problem dependent and change with time. Numerical examples show that the results of our scheme compare well with other methods for one and two-dimensional problems. In particular, the algorithm can capture well complex flow features of the one-dimensional shock entropy wave interaction problem and two-dimensional shock–bubble interaction problem.     

Geometric conservation law and applications to high-order finite difference schemes with stationary grids

The geometric conservation law (GCL) includes the volume conservation law (VCL) and the surface conservation law (SCL). Though the VCL is widely discussed for time-depending grids, in the cases of stationary grids the SCL also works as a very important role for high-order accurate numerical simulations. The SCL is usually not satisfied on discretized grid meshes because of discretization errors, and the violation of the SCL can lead to numerical instabilities especially when high-order schemes are applied. In order to fulfill the SCL in high-order finite difference schemes, a conservative metric method (CMM) is presented. This method is achieved by computing grid metric derivatives through a conservative form with the same scheme applied for fluxes. The CMM is proven to be a sufficient condition for the SCL, and can ensure the SCL for interior schemes as well as boundary and near boundary schemes. Though the first-level difference operators δ₃ have no effects on the SCL, no extra errors can be introduced as δ₃ = δ₂. The generally used high-order finite difference schemes are categorized as central schemes (CS) and upwind schemes (UPW) based on the difference operator δ₁ which are used to solve the governing equations. The CMM can be applied to CS and is difficult to be satisfied by UPW. Thus, it is critical to select the difference operator δ₁ to reduce the SCL-related errors. Numerical tests based on WCNS-E-5 show that the SCL plays a very important role in ensuring free-stream conservation, suppressing numerical oscillations, and enhancing the robustness of the high-order scheme in complex grids.     

A Riemann solver for unsteady computation of 2D shallow flows with variable density

A novel 2D numerical model for vertically homogeneous shallow flows with variable horizontal density is presented. Density varies according to the volumetric concentration of different components or species that can represent suspended material or dissolved solutes. The system of equations is formed by the 2D equations for mass and momentum of the mixture, supplemented by equations for the mass or volume fraction of the mixture constituents. A new formulation of the Roe-type scheme including density variation is defined to solve the system on two-dimensional meshes. By using an augmented Riemann solver, the numerical scheme is defined properly including the presence of source terms involving reaction. The numerical scheme is validated using analytical steady-state solutions of variable-density flows and exact solutions for the particular case of initial value Riemann problems with variable bed level and reaction terms. Also, a 2D case that includes interaction with obstacles illustrates the stability and robustness of the numerical scheme in presence of non-uniform bed topography and wetting/drying fronts. The obtained results point out that the new method is able to predict faithfully the overall behavior of the solution and of any type of waves.     

解流体力学方程组的一种隐式完全守恒差分格式

对Lagrange非守恒流体力学方程组给出了一种隐式完全守恒差分格式,既保证了质量、动量和总能量守恒的差分近似,又能满足内能与动能的平衡特性,提高了数值解的精度。并用该格式对两个可压缩理想流体模型进行了数值计算,并与其它差分格式作了比较。     

求解多维欧拉方程的二阶非结构网格混合旋转Riemann求解器

将基于旋转近似Riemann求解器的二阶精度迎风型有限体积方法推广到非结构网格,采用基于网格中心的有限体积法,梯度的计算采用基于节点的方法引入更多的控制体模板,限制器的构造采用与非结构化网格相适应的形式.在求解Riemann问题时,沿具有一定物理意义的两个迎风方向,即控制体界面两侧速度差矢量方向及与之正交的方向.能够完全消除基于Riemann求解器的通量差分裂格式存在的激波不稳定或"红斑"现象.为减小计算量,采用HLL和Roe FDS混合旋转格式.     

A new formulation of Kapila’s five-equation model for compressible two-fluid flow,and its numerical treatment

A new formulation of Kapila’s five-equation model for inviscid, non-heat-conducting, compressible two-fluid flow is derived, together with an appropriate numerical method. The new formulation uses flow equations based on conservation laws and exchange laws only. The two fluids exchange momentum and energy, for which exchange terms are derived from physical laws. All equations are written as a single system of equations in integral form. No equation is used to describe the topology of the two-fluid flow. Relations for the Riemann invariants of the governing equations are derived, and used in the construction of an Osher-type approximate Riemann solver. A consistent finite-volume discretization of the exchange terms is proposed. The exchange terms have distinct contributions in the cell interior and at the cell faces. For the exchange-term evaluation at the cell faces, the same Riemann solver as used for the flux evaluation is exploited. Numerical results are presented for two-fluid shock-tube and shock-bubble-interaction problems, the former also for a two-fluid mixture case. All results show good resemblance with reference results.     

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.     

A new class of gas-kinetic relaxation schemes for the compressible Euler equations

Starting from the gas-kinetic model, a new class of relaxation schemes for the Euler equations is presented. In contrast to the Riemann solver, these schemes provide a multidimensional dynamical gas evolution model, which combines both Lax-Wendroff and kinetic flux vector splitting schemes, and their coupling is based on the fact that a nonequilibrium state will evolve into an equilibrium state along with the increase of entropy. The numerical fluxes are constructed without getting into the details of the particle collisions. The results for many well-defined test cases are presented to indicate the robustness and accuracy of the current scheme.