Gas kinetic flux solver based finite volume weighted essentially non-oscillatory scheme for inviscid compressible flows

A high-order gas kinetic flux solver(GKFS)is presented for simulating inviscid compressible flows.The weighted essentially non-oscillatory(WENO)scheme on a uniform mesh in the finite volume formulation is combined with the circular function-based GKFS(C-GKFS)to capture more details of the flow fields with fewer grids.Different from most of the current GKFSs,which are constructed based on the Maxwellian distribution function or its equivalent form,the C-GKFS simplifies the Maxwellian distribution function into the circular function,which ensures that the Euler or Navier-Stokes equations can be recovered correctly.This improves the efficiency of the GKFS and reduces its complexity to facilitate the practical application of engineering.Several benchmark cases are simulated,and good agreement can be obtained in comparison with the references,which demonstrates that the high-order C-GKFS can achieve the desired accuracy.

Interface flux reconstruction method based on optimized weight essentially non-oscillatory scheme

Aimed at the computational aeroacoustics multi-scale problem of complex configurations discretized with multi-size mesh, the flux reconstruction method based on modified Weight Essentially Non-Oscillatory（WENO） scheme is proposed at the interfaces of multi-block grids.With the idea of Dispersion-Relation-Preserving（DRP） scheme, different weight coefficients are obtained by optimization, so that it is in WENO schemes with various characteristics of dispersion and dissipation. On the basis, hybrid flux vector splitting method is utilized to intelligently judge the amplitude of the gap between grid interfaces. After the simulation and analysis of 1D convection equation with different initial conditions, modified WENO scheme is proved to be able to independently distinguish the gap amplitude and generate corresponding dissipation according to the grid resolution. Using the idea of flux reconstruction at grid interfaces, modified WENO scheme with increasing dissipation is applied at grid points, while DRP scheme with low dispersion and dissipation is applied at the inner part of grids. Moreover, Gauss impulse spread and periodic point sound source flow among three cylinders with multi-scale grids are carried out. The results show that the flux reconstruction method at grid interfaces is capable of dealing with Computational Aero Acoustics（CAA） multi-scale problems.

Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

Hypersonic Shock Wave/Boundary Layer Interactions by a Third-Order Optimized Symmetric WENO Scheme

A novel third-order optimized symmetric weighted essentially non-oscillatory(WENO-OS3)scheme is used to simulate the hypersonic shock wave/boundary layer interactions.Firstly,the scheme is presented with the achievement of low dissipation in smooth region and robust shock-capturing capabilities in discontinuities.The Maxwell slip boundary conditions are employed to consider the rarefied effect near the surface.Secondly,several validating tests are given to show the good resolution of the WENO-OS3 scheme and the feasibility of the Maxwell slip boundary conditions.Finally,hypersonic flows around the hollow cylinder truncated flare(HCTF)and the25°/55°sharp double cone are studied.Discussions are made on the characteristics of the hypersonic shock wave/boundary layer interactions with and without the consideration of the slip effect.The results indicate that the scheme has a good capability in predicting heat transfer with a high resolution for describing fluid structures.With the slip boundary conditions,the separation region at the corner is smaller and the prediction is more accurate than that with no-slip boundary conditions.

Quinpi:Integrating Conservation Laws with CWENO Implicit Methods

Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws.

Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

爆轰波模拟中一个保正的有限体积WENO格式

对一维爆轰波的数值模拟设计了一种保正的有限体积WENO格式.以一维反应欧拉方程组作为描述爆轰波的控制方程,对方程组在空间离散上采用三阶WENO重构的有限体积法,时间离散上采用Strang分裂法和二阶龙格库塔法.从爆轰波的数值模拟中可以观察到,在压力快速变化的区域使用一般的WENO重构方法会使得压力出现负值.提出了一种简单且有效的策略,使得重构的压力具有保正性.通过数值算例验证了所提出的数值格式的稳定性和收敛性,以及对爆轰波结构变化捕捉的良好能力.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.

Adaptive OrderWENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order.This is an extension of the conservative semi-Lagrangian(SL)finite difference WENO scheme in[Qiu and Shu,JCP,230(4)(2011),pp.863-889],in which linear weights in SL WENO framework were shown to not exist for variable coefficient problems.Hence,the order of accuracy is not optimal from reconstruction stencils.In this paper,we incorporate a recent WENO adaptive order(AO)technique[Balsara et al.,JCP,326(2016),pp.780-804]to the SL WENO framework.The new scheme can achieve an optimal high order of accuracy,while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO.The positivity-preserving limiter is further applied to ensure the positivity of solutions.Finally,the scheme is applied to high dimensional problems by a fourth-order dimensional splitting.We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations,the Vlasov-Poisson system,the guiding center Vlasov model as well as the incompressible Euler equations.

On the Positivity of Linear Weights in WENO Approximations

High order accurate weighted essentially non-oscillatory （WENO） schemes have been used extensively in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation to first and second derivatives, and integration procedures.

Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave

A hyperbolic conservation equation can easily generate strong discontinuous solutions such as shock waves and contact discontinuity.By introducing the arc-length parameter,the pseudo arc-length method(PALM)smoothens the discontinuous solution in the arc-length space.This in turn weakens the singularity of the equation.To avoid constructing a high-order scheme directly in the deformed physical space,the entire calculation process is conducted in a uniform orthogonal arc-length space.Furthermore,to ensure the stability of the equation,the time step is reduced by limiting the moving speed of the mesh.Given that the calculation does not involve the interpolation process of physical quantities after the mesh moves,it maintains a high computational efficiency.The numerical examples show that the algorithm can effectively reduce numerical oscillations while maintaining excellent characteristics such as high precision and high resolution.

Heterogeneous Parallel Algorithm Design and Performance Optimization for WENO on the Sunway TaihuLight Supercomputer

A Weighted Essentially Non-Oscillatory scheme(WENO) is a solution to hyperbolic conservation laws,suitable for solving high-density fluid interface instability with strong intermittency. These problems have a large and complex flow structure. To fully utilize the computing power of High Performance Computing(HPC) systems, it is necessary to develop specific methodologies to optimize the performance of applications based on the particular system’s architecture. The Sunway TaihuLight supercomputer is currently ranked as the fastest supercomputer in the world. This article presents a heterogeneous parallel algorithm design and performance optimization of a high-order WENO on Sunway TaihuLight. We analyzed characteristics of kernel functions, and proposed an appropriate heterogeneous parallel model. We also figured out the best division strategy for computing tasks,and implemented the parallel algorithm on Sunway TaihuLight. By using access optimization, data dependency elimination, and vectorization optimization, our parallel algorithm can achieve up to 172× speedup on one single node, and additional 58× speedup on 64 nodes, with nearly linear scalability.

Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes

We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.

On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory(WENO)schemes in two dimensional Cartesian meshes.We compare them in terms of accuracy,performance for smooth and shocked solutions,and efficiency in CPU timing.For linear systems both schemes are high order accurate,however for nonlinear systems,analysis and numerical simulation results verify that one of them(Class A)is only second order accurate,while the other(Class B)is high order accurate.The WENO scheme in Class A is easier to implement and costs less than that in Class B.Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes,despite of the difference in their formal order of accuracy.The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

A robust WENO scheme for nonlinear waves in a moving reference frame

For robust nonlinear wave simulation in a moving reference frame, we recast the free surface problem in Hamilton-Jacobi form and propose a Weighted Essentially Non-Oscillatory （WENO） scheme to automatically handle the upwinding of the convective term. A new automatic procedure for deriving the linear WENO weights based on a Taylor series expansion is introduced. A simplified smoothness indicator is proposed and is shown to perform well. The scheme is combined with high-order explicit Runge-Kutta time integration and a dissipative Lax-Friedrichs-type flux to solve for nonlinear wave propagation in a moving frame of reference. The WENO scheme is robust and less dissipative than the equivalent order upwind-biased finite difference scheme for all ratios of frame of reference to wave propagation speed tested. This provides the basis for solving general nonlinear wave-structure interaction problems at forward speed.

High Order Finite Difference Hermite WENO Fixed-Point Fast Sweeping Method for Static Hamilton-Jacobi Equations

In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any HamiltonJacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.

基于小波奇异分析的流场计算方法及应用

首次将三维小波以及二维小波奇异性分析的思想引进三维以及二维复杂流场的数值计算,发展了一种高效率、高精度、高分辨率的方法.该算法的核心是获取流场中物理量在不同空间点的Hlder指数α,而该指数α的获取又依赖于小波变换以及高维（即二维或三维）小波分析技术.在三维与二维欧氏空间中,为进行小波多分辨分析,需要在尺度空间与小波空间分别引进尺度基与小波基.对二维问题,尺度基与小波基的基底要由1个尺度函数与3个小波函数组成,而三维时要由1个尺度函数和7个小波函数组成.借助于小波奇异分析所找到的流场中奇异点区域与光滑区域,便可分别选用高分辨率、高精度的WENO（weighted essen-tial non-oscillatory）格式与高精度中心差分格式进行流场的离散求解.一系列二维（即,①二维前台阶问题的Euler流;②二维双马赫反射的Euler流;③著名的二维Riemann问题;④跨声速RAE2822翼型二维绕流;⑤跨声速VKI-LS 59二维涡轮叶栅绕流）与三维（即,⑥跨声速轴流压气机转子NASA（National Aeronauticsand Space Administration）Rotor 37三维黏性绕流;⑦跨声速风扇转子NASA Rotor 67三维黏性绕流）算例表明：该方法的计算效率比传统的WENO格式有较大的提高,大部分典型算例能够提高3~5倍,而且可以获取复杂流场中高分辨率的激波结构以及涡系分布,可以得到与有关实验数据较为吻合的数值结果.

低雷诺数NACA0012平面叶栅流场直接数值模拟

采用具有7阶精度的weighted essentially non-oscillatory(WENO)差分格式,直接求解可压缩二维非定常N-S方程组,研究了NACA0012翼型平面叶栅低雷诺数流动的特征.直接模拟及与文献对比的结果表明:叶栅尾缘涡脱落的形成过程与圆柱绕流涡脱落的形成过程非常相似.平面叶栅尾迹区的2阶统计量与孤立翼型尾迹区的2阶统计量具有相同的分布特征,但前者的强度显著大于后者.周期性的涡脱落不仅在上下翼面形成非定常分离,也使得尾迹区某点的总压发生准周期性的变化.随着栅距的减小,翼型上的平均分离位置向前缘移动;尾迹区某点的总压变化频率及其幅值均显著地增加;而且栅距越小,速度脉动2阶统计量反而越大.

Numerical solutions of a multi-class traffic flow model on an inhomogeneous highway using a high-resolution relaxed scheme

A high-resolution relaxed scheme which requires little information of the eigenstructure is presented for the multiclass Lighthill-Whitham-Richards （LWR） model on an inhomogeneous highway. The scheme needs only an estimate of the upper boundary of the maximum of absolute eigenvalues. It is based on incorporating an improved fifth-order weighted essentially non-oscillatory （WENO） reconstruction with relaxation approximation. The scheme benefits from the simplicity of relaxed schemes in that it requires no exact or approximate Riemann solvers and no projection along characteristic directions. The effectiveness of our method is demonstrated in several numerical examples.