High-order targeted essentially non-oscillatory scheme for two-fluid plasma model

The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.

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.

Third-order modified coeffcient scheme based on essentially non-oscillatory scheme

A third-order numerical scheme is presented to give approximate solutions to multi-dimensional hyperbolic conservation laws only using modified coefficients of an essentially non-oscillatory （MCENO） scheme without increasing the base points during construction of the scheme. The construction process shows that the modified coefficient approach preserves favourable properties inherent in the original essentially nonoscillatory （ENO） scheme for its essential non-oscillation, total variation bounded （TVB）, etc. The new scheme improves accuracy by one order compared to the original one. The proposed MCENO scheme is applied to simulate two-dimensional Rayleigh-Taylor （RT） instability with densities 1：3 and 1：100, and solve the Lax shock-wave tube numerically. The ratio of CPU time used to implement MCENO, the .third-order ENO and fifth-order weighed ENO （WENO） schemes is 0.62：1：2.19. This indicates that MCENO improves accuracy in smooth regions and has higher accuracy and better efficiency compared to the original ENO scheme.

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.

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.

Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory(WENO)finite difference schemes.Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems,and the twodimensional Rayleigh-Taylor instability(RTI)problem as examples,we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition,which are often given and used in the literature,and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem.An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error.The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability,reduced rounding errors and increased computational efficiency.

EXISTENCE OF NON-OSCILLATORY SOLUTIONS OF A CLASS OF HIGHER-ORDER NONLINEAR NEUTRAL DIFFERENCE EQUATIONS WITH A FORCED TERM

In this paper, we consider a certain forced higher-order nonlinear neutral difference equation. By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain necessary and sufficient conditions for the existence of non-oscillatory solutions for the system. Our results extend the results of Zhang [5].

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.

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.

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.

New conception in continuum theory of constitutive equation for anisotropic crystalline polymer liquids

A new continuum theory of the constitutive equation of co-rotational derivative type is developed for anisotropic viscoelastic fluid—liquid crystalline (LC) polymers. A new concept of simple anisotropic fluid is introduced. On the basis of principles of anisotropic simple fluid, stress behaviour is described by velocity gradient tensor and spin tensor instead of the velocity gradient tensor in the classic Leslie—Ericksen continuum theory. Analyzing rheological nature of the fluid and using tensor analysis a general form of the constitutive equ- ation of co-rotational type is established for the fluid. A special term of high order in the equation is introduced by author to describe the sp- ecial change of the normal stress differences which is considered as a result of director tumbling by Larson et al. Analyzing the experimental results by Larson et al., a principle of Non- oscillatory normal stress is introduced which leads to simplification of the problem with relaxation times. The special behaviour of non- symmetry of the shear stress is predicted by using the present model for LC polymer liquids. Two shear stresses in shear flow of LC polymer liquids may lead to vortex and rotation flow, i.e. director tumbling in the flow. The first and second normal stress differences are calculated by the model special behaviour of which is in agree- ment with experiments. In the research, the com- putational symbolic manipulation such as computer software Maple is used. For the anisotropic viscoelastic fluid the constitutive equation theory is of important fundamental significance.

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.

DEVELOPMENT AND APPLICATIONS OF WENO SCHEMES IN CONTINUUM PHYSICS

This paper briefly presents the general ideas of high order accurate weighted essentially non oscillatory (WENO) schemes, and describes the similarities and differences of the two classes of WENO schemes: finite vo lume schemes and finite difference schemes. We also briefly mention a recent development of WENO schemes, namely an adaptive approach within the finite difference framework using smooth time dependent curvilinear coordinates.

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory （WENO） scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws （MUSCL）. For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the struc- tured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the so- lution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.

On the contact width in generalized plain strain JKR model for isotropic solids

Adhesive contact model between an elastic cylinder and an elastic half space is studied in the present paper, in which an external pulling force is acted on the above cylinder with an arbitrary direction and the contact width is assumed to be asymmetric with respect to the structure. Solutions to the asymmetric model are obtained and the effect of the asymmetric contact width on the whole pulling process is mainly discussed. It is found that the smaller the absolute value of Dundurs＇ parameter 13 or the larger the pulling angle O, the more reasonable the symmetric model would be to approximate the asymmetric one.

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.

APPLICATION OF WEIGHTED NONOSCILLATORY AND NON-FREE-PARAMETER DISSIPATION DIFFERENCE SCHEME IN CALCULATING THE FLOW OF VIBRATING FLAT CASCADE

A dual-time method is introduced to calculate the unsteady flow in a certain vibrating flat cascade. An implicit lower-upper symmetric-gauss-seidel scheme（LU-SGS） is applied for time stepping in pseudo time domains, and the convection items are discretized with the spatial three-order weighted non-oscillatory and non-free-parameter dissipation difference （WNND） scheme. The turbulence model adopts q-co low-Reynolds-number model. The frequency specmuns of lift coefficients and the unsteady pressure-difference coefficients at different spanwise heights as well as the entropy contours at blade tips on different vibrating instants, are obtained. By the analysis of frequency specmuns of lift coefficients at three spanwise heights, it is considered that there exist obvious non-linear perturbations in the flow induced by the vibrating, and the perturbation frequencies are higher than the basic frequency. The entropy contours at blade tips at different times display an intensively unsteady attribute of the flow under large amplitudes.

HIGH ORDER FINITE DIFFERENCE HERMITE WENO FAST SWEEPING METHODS FOR STATIC HAMILTON-JACOBI EQUATIONS

In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.

Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models

Non-equilibrium hyperbolic traffic models can be derived as continuum approximations of car-following models and in many cases the resulting continuum models are non-conservative.This leads to numerical difficulties,which seem to have discouraged further development of complex behavioral continuum models,which is a significant research need.In this paper,we develop a robust numerical scheme that solves hyperbolic traffic flow models based on their non-conservative form.We develop a fifth-order alternative weighted essentially non-oscillatory(A-WENO)finite-difference scheme based on the path-conservative central-upwind(PCCU)method for several non-equilibrium traffic flow models.In order to treat the non-conservative product terms,we use a path-conservative technique.To this end,we first apply the recently proposed secondorder finite-volume PCCU scheme to the traffic flow models,and then extend this scheme to the fifth-order of accuracy via the finite-difference A-WENO framework.The designed schemes are applied to three different traffic flow models and tested on a number of challenging numerical examples.Both schemes produce quite accurate results though the resolution achieved by the fifth-order A-WENO scheme is higher.The proposed scheme in this paper sets the stage for developing more robust and complex continuum traffic flow models with respect to human psychological factors.