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.

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.

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.

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.

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.

High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations

In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.

R-T instability model of magnetic fluid and its numerical simulations

The Rayleigh-Taylor(R-T) instability of ferrofluid has been the subject of recent research,because of its implications on the stability of stellar.By neglecting the viscosity and rotation of magnetic fluid,and assuming that the magnetic particles are irrotational and temperature insensitive,we obtain a simplified R-T instability model of magnetic fluid.For the interface tracing,we use five-order weighted essentially non-oscillatory(WENO) scheme to spatial direction and three-order TVD R-K method to time direction on the uniform mesh,respectively.If the direction of the external magnetic field is the same as that of gravity,the velocities of the interface will be increased.But if the direction of the external magnetic field is in opposition to the direction of gravity,the velocities of the interface will be decreased.When the direction of the external magnetic field is perpendicular to the direction of gravity,the symmetry of the interface will be destroyed.Because of the action which is produced by perpendicular external magnetic field,there are other bubbles at the boudaries which parallel the direction of gravity.When we increase the magnetic susceptibility of the magnetic fluids,the effects of external magnetic fields will be more distinct for the interface tracing.

The Formulation of Finite Difference RBFWENO Schemes for Hyperbolic Conservation Laws:An Alternative Technique

To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical WENO-JS weights,a new type of WENO schemes has been proposed to solve conservation laws[J.Guo et al.,J.Sci.Comput.,70(2017),pp.551–575].The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws.Unlike the usual method for reconstructing the flux functions,the flux function is generated directly with the conservative variables.Comparing with Guo and Jung(2017),the main advantage of this framework is that arbitrary monotone fluxes can be employed,while in Guo and Jung(2017)only smooth flux splitting can be used to reconstruct flux functions.Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.

A CHARACTERISTIC-BASED FINITE VOLUME SCHEME FOR SHALLOW WATER EQUATIONS

We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We apply the scheme to simulate dam-break problems. A number of challenging test cases are considered, such as large depth differences even wet/dry bed. The numerical solutions well agree with the analytical solutions. The results demonstrate the desired accuracy, high-resolution and robustness of the presented scheme.

A High Resolution Low Dissipation Hybrid Scheme for Compressible Flows

In this paper, an efficient hybrid shock capturing scheme is proposed to obtain accurate results both in the smooth region and around discontinuities for compressible flows. The hybrid algorithm is based on a fifth-order weighted essentially non-oscillatory （WENO） scheme in the finite volume form to solve the smooth part of the flow field, which is coupled with a characteristic-based monotone upstream-centered scheme for conservation laws （MUSCL） to capture discontinuities. The hybrid scheme is intended to combine high resolution of MUSCL scheme and low dissipation of WENO scheme. The two ingredients in this hybrid scheme are switched with an indicator. Three typical indicators are chosen and compared. MUSCL and WENO are both shock capturing schemes making the choice of the indicator parameter less crucial. Several test cases are carried out to investigate hybrid scheme with different indicators in terms of accuracy and efficiency. Numerical results demonstrate that the hybrid scheme in the present work performs well in a broad range of problems.

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 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.

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.

Simulations of Shallow Water Equations by Finite Difference WENO Schemes with Multilevel Time Discretization

In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory(WENO)schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms.Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values.Extensive simulations are performed,which indicate that,the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy,and are more cost effective than WENO scheme with Runge-Kutta time discretization,while still maintaining nonoscillatory properties.

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.

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper,we develop twofinite difference weighted essentially non-oscillatory(WENO)schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi(DP)andµ-Degasperis-Procesi(µDP)equations,which contain nonlinear high order derivatives,and possibly peakon solutions or shock waves.By introducing auxiliary variable(s),we rewrite the DP equation as a hyperbolic-elliptic system,and theµDP equation as afirst order system.Then we choose a linearfinite difference scheme with suitable order of accuracy for the auxiliary variable(s),and twofinite difference WENO schemes with unequal-sized sub-stencils for the primal variable.One WENO scheme uses one large stencil and several smaller stencils,and the other WENO scheme is based on the multi-resolution framework which uses a se-ries of unequal-sized hierarchical central stencils.Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil,both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights.Another advantage is that thefinal reconstructed polynomial on the target cell is a polynomial of the same de-gree as the polynomial over the big stencil,while the classicalfinite difference WENO reconstruction can only be obtained for specific points inside the target interval.Nu-merical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

HIGH ORDER STABLE MULTI-DOMAIN HYBRID RKDG AND WENO-FD METHODS

Recently, a kind of high order hybrid methods based on Runge-Kutta discontinu- ous Galerkin （RKDG） method and weighted essentially non-oscillatory finite difference （WENO-FD） scheme was proposed. Those methods are computationally efficient, however stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be ＇downstream＇. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be ＇upstream＇. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing （TVD） condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.