A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

A New Hybrid WENO Scheme with the High-Frequency Region for Hyperbolic Conservation Laws

In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridization,the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region;meanwhile,the essentially oscillation-free solution could also be obtained.By adapting the original smoothness indicator in the WENO reconstruction,the stencil is distinguished into three types:smooth,non-smooth,and high-frequency region.In the smooth region,the linear reconstruction is used and the non-smooth region with the WENO reconstruction.In the high-frequency region,the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor,which could capture high-frequency wave efficiently.Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.

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.

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.

An efficient unstructured WENO method for supersonic reactive flows

An efficient high-order numerical method for supersonic reactive flows is proposed in this article.The reactive source term and convection term are solved separately by splitting scheme.In the reaction step,an adaptive time-step method is presented,which can improve the efficiency greatly.In the convection step,a third-order accurate weighted essentially non-oscillatory（WENO）method is adopted to reconstruct the solution in the unstructured grids.Numerical results show that our new method can capture the correct propagation speed of the detonation wave exactly even in coarse grids,while high order accuracy can be achieved in the smooth region.In addition,the proposed adaptive splitting method can reduce the computational cost greatly compared with the traditional splitting method.

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the one- dimensional scalar and inviscid Euler conservation laws. The Simpson＇s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson＇s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory （CWENO） reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

In this paper,we propose a new conservative semi-Lagrangian(SL)finite difference(FD)WENO scheme for linear advection equations,which can serve as a base scheme for the Vlasov equation by Strang splitting[4].The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume(FV)WENO scheme[3].However,instead of inputting cell averages and approximate the integral form of the equation in a FV scheme,we input point values and approximate the differential form of equation in a FD spirit,yet retaining very high order(fifth order in our experiment)spatial accuracy.The advantage of using point values,rather than cell averages,is to avoid the second order spatial error,due to the shearing in velocity(v)and electrical field(E)over a cell when performing the Strang splitting to the Vlasov equation.As a result,the proposed scheme has very high spatial accuracy,compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson(VP)system.We perform numerical experiments on linear advection,rigid body rotation problem;and on the Landau damping and two-stream instabilities by solving the VP system.For comparison,we also apply(1)the conservative SL FD WENO scheme,proposed in[22]for incompressible advection problem,(2)the conservative SL FD WENO scheme proposed in[21]and(3)the non-conservative version of the SL FD WENO scheme in[3]to the same test problems.The performances of different schemes are compared by the error table,solution resolution of sharp interface,and by tracking the conservation of physical norms,energies and entropies,which should be physically preserved.

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.

A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids

A recent work of Li et al.[Numer.Math.Theor.Meth.Appl.,1(2008),pp.92-112]proposed a finite volume solver to solve 2D steady Euler equations.Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region,the overshoot or undershoot phenomenon can still be observed.Moreover,the numerical accuracy is degraded by using Venkatakrishnan limiter.To fix the problems,in this paper the WENO type reconstruction is employed to gain both the accurate approximations in smooth region and non-oscillatory sharp profiles near the shock discontinuity.The numerical experiments will demonstrate the efficiency and robustness of the proposed numerical strategy.

On the Solution Accuracy Downstream of Shocks When Using Godunov-Type Schemes.I.Sources of Errors in One-Dimensional Problems

The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity approach.The proposed paper describes the numerical methods used in the study,and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems.The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.

High-order compact gas-kinetic schemes for three-dimensional flow simulations on tetrahedral mesh

A general framework for the development of high-order compact schemes has been proposed recently.The core steps of the schemes are composed of the following.1).Based on a kinetic model equation,from a generalized initial distribution of flow variables construct a time-accurate evolution solution of gas distribution function at a cell interface and obtain the corresponding flux function;2).Introduce the WENO-type weighting functions into the high-order time-derivative of the cell interface flux function in the multistage multi-derivative(MSMD)time stepping scheme to cope with the possible impingement of a shock wave on a cell interface within a time step,and update the cell-averaged conservative flow variables inside each control volume;3).Model the time evolution of the gas distribution function on both sides of a cell interface separately,take moments of the inner cell interface gas distribution function to get flow variables,and update the cell-averaged gradients of flow variables inside each control volume;4).Based on the cell-averaged flow variables and their gradients,develop compact initial data reconstruction to get initial condition of flow distributions at the beginning of next time step.A compact gas-kinetic scheme(GKS)up to sixth-order accuracy in space and fourth-order in time has been constructed on 2D unstructured mesh.In this paper,the compact GKS up to fourth-order accuracy on three-dimensional tetrahedral mesh will be further constructed with the focus on the WENO-type initial compact data reconstruction.Nonlinear weights are designed to achieve high-order accuracy for the smooth Navier-Stokes solution and keep super robustness in 3D computation with strong shock interactions.The fourth-order compact GKS uses a large time step with a CFL number 0.6 in the simulations from subsonic to hypersonic flow.A series of test cases are used to validate the scheme.The high-order compact GKS can be used in 3D applications with complex geometry.

A high-order multidimensional gas-kinetic scheme for hydrodynamic equations

This paper concerns the development of high-order multidimensional gas kinetic schemes for the Navier-Stokes solutions.In the current approach,the state-of-the-art WENO-type initial reconstruction and the gas-kinetic evolution model are used in the construction of the scheme.In order to distinguish the physical and numerical requirements to recover a physical solution in a discretized space,two particle collision times will be used in the current high-order gas-kinetic scheme(GKS).Different from the low order gas dynamic model of the Riemann solution in the Godunov type schemes,the current method is based on a high-order multidimensional gas evolution model,where the space and time variation of a gas distribution function along a cell interface from an initial piecewise discontinuous polynomial is fully used in the flux evaluation.The high-order flux function becomes a unification of the upwind and central difference schemes.The current study demonstrates that both the high-order initial reconstruction and high-order gas evolution model are important in the design of a high-order numerical scheme.Especially,for a compact method,the use of a high-order local evolution solution in both space and time may become even more important,because a short stencil and local low order dynamic evolution model,i.e.,the Riemann solution,are contradictory,where valid mechanism for the update of additional degrees of freedom becomes limited.

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike resolution will be presented.Based on the high-order gas evolution model,both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface.As a result,inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step.The flow variable update and slope update are coming from the same physical solution at the cell interface.This strategy needs time accurate solution at a cell interface,which cannot be achieved by the Riemann problem based flow solvers,even though they can also provide the interface flux functions and interface flow variables.Instead,in order to update the slopes in the Riemann-solver based schemes,such as HWENO,there are additional governing equations for slopes or equivalent degrees of freedom inside each cell.In GKS,only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables.Based on both cell averaged values and their slopes,compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed.As analyzed in this paper,the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives.For nonlinear gas dynamic evolution,in order to avoid spurious oscillation in discontinuous region,the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction.Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction.In GKS,the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation,which covers a physical process from an initial non-equilibrium state to a final equilibrium one.Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction,and the equilibrium state is basically constructed from the linear symmetric reconstruction,the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function.This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions.This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions.Compared with discontinuous Galerkin(DG)scheme,the current compact GKS uses the same local and compact stencil,achieves the 6th-order and 8th-order accuracy,uses a much larger time step with CFL number≥0.3,has the robustness as a 2nd-order scheme,and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process.The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique.At the same time,the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface.Due to the use of multi-stage multi-derivative(MSMD)time-stepping technique,for achieving a 4th-order time accuracy,the current scheme uses only two stages instead of four in the traditional Runge-Kutta method.As a result,the current GKS becomes much more efficient than the corresponding same order DG method.A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS.The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions,i.e.,strong shock discontinuity,shear instability,aero-acoustic wave propagation,and NS solutions.It promotes the development of high-order scheme to a new level of maturity.The success of the current scheme crucially depends on the high-order gas evolution model,which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.

A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations

Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics.The main purpose of this work is to demonstrate our claim in[3]that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges,which restricts the accuracy of the resulting scheme to at most second order.The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes.Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.

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.