Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

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.

New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

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.

采用重组模板的权重优化WENO-Z格式

针对精确模拟含激波等复杂流动结构的流场对高精度格式的低耗散低色散要求,基于5阶有限差分WENO-Z格式,提出一种模板重组技术。在计算WENO非线性权时,引入一个由3点模板重新组合的4点模板,优化原格式中各模板的权重分配,进而提出了两种改进WENO-Z格式。采用近似色散关系分析方法对改进前后格式色散与耗散特性进行了对比与分析。分析表明:两种改进格式耗散有不同程度的降低。数值实验表明:改进格式具有更优越的激波捕捉性能,对小尺度流场结构具有更高的分辨率。

High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.

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.

求解双曲守恒律方程的高阶WENO型熵相容格式的对比研究

本文采用五阶WENO重构方法,结合熵相容格式,得到求解双曲守恒律方程的高阶WENO型熵相容格式.数值实验证明了所得格式的鲁棒性.

基于混合模板改进的五阶WENO格式在求解双曲守恒律方程中的应用

本文在WENO-MZQ3格式的基础上,将其模板推广到五阶WENO,即在经典五阶WENO格式的三个三点模板的基础上引入五点模板,并引入新的改进的全局光滑因子,对其进行加权处理得到新的五阶WENO格式.理论分析表明新的五阶WENO格式在光滑区域比五阶WENO-JS格式具有更高阶的计算精度,通过一系列一维和二维数值算例也验证了新的五阶WENO格式具有更高的精度和分辨率.

Cavitating/non-cavitating flows simulation by third-order finite volume scheme and power-law preconditioning method

Equations of steady inviscid and laminar flows are solved by means of a third-order finite volume （FV） scheme. For this purpose, a cell-centered discretization technique is employed. In this technique, the flow parameters at the cell faces are computed using a third-order weighted averages procedure. A fourth-order artificial dissipation is used for stability of the solution. In order to achieve the steady-state situation, four-step Runge-Kutta explicit time integration method is applied. An advanced progressive preconditioning method, named the power-law preconditioning method, is used for faster convergence. In this method, the preconditioning matrix is adjusted automatically from the velocity and/or pressure flow-field by a power-law relation. Attention is directed towards accuracy and convergence of the schemes. The results presented in the paper focus on steady inviscid and laminar flows around sheet-cavitating and fully-wetted bodies including hydrofoils and circular/elliptical cylinder. Excellent agreements are obtained when numerical predictions are compared with other available experimental and numerical results. In addition, it is found that using the power-law preconditioner significantly increases the numerical convergence speed.

一种基于欧拉方程通量分裂的五阶有限差分共权多分辨率WENO格式

本文研究了欧拉方程的高精度数值解法,在多分辨率WENO数值格式中引入了共权思想,获得了一种新的五阶有限差分共权多分辨率WENO格式.数值实验表明,该方法在解的光滑区域能获得一致的高精度,在强间断附近能保持数值解基本无振荡的性质,从而验证算法的有效性.

非均匀结构网格上MUSCL和WENO格式的精度

基于一维均匀网格条件下构造的差分格式,在实际应用中须推广到非均匀或者曲线网格上,坐标变换过程引入几何诱导误差。目前常用收敛解误差随着网格细化变化的精度测试方法评估差分格式的精度。在二维柱坐标均匀网格上,采用1阶迎风、2阶MUSCL和5阶WENO计算流场参数为常数的自由流问题,按照精度测试方法比较收敛曲线斜率,发现1阶迎风的网格收敛精度是2阶的,5阶WENO的网格收敛精度不到1阶。理论分析表明,这种精度测试方法与差分格式精度定义不等价,而且所采用的数据无法反映差分格式的固有缺陷,因此,不能用来作为差分格式精度评价指标。很多研究WENO的文献经常模拟双Mach反射问题、二维Riemann问题等经典算例,把接触间断是否演变成不稳定涡结构作为特征,理论上可以证明涡结构是非物理现象,因此用是否出现涡结构作为算法高精度的论据并不合适。

7阶WENO-S格式的计算效率研究

WENO-S格式是一类适合于含间断问题数值模拟的加权本质无振荡格式.这类格式的光滑因子满足对单频波为常数,这使得其近似色散关系与线性基底格式一致,并且具有良好的小尺度波动模拟能力.计算效率是数值方法性能指标的一个重要方面.由于WENO-S格式的光滑因子在各子模板上的计算公式除下标不同外形式一致,在计算线性对流方程相邻数值通量时,部分光滑因子完全相同.为此提出一种消除WENO-S格式冗余光滑因子计算的方法.该方法要求一条网格线上用于重构或插值的量可以表示为一个序列.基于此要求分析其对于几种不同物理问题的可行性和使用方法.以7阶WENO-S格式为例介绍了格式性质和去除冗余光滑因子计算的方法.该方法中预先计算和存储一条网格线上的所有光滑因子,在网格点较多的情况下,光滑因子计算次数约为原7阶WENO-S格式的1/4.对一维对流问题、球面波传播问题、二维旋转问题、二维小扰动传播问题及一维和二维无黏流动问题进行了数值模拟.结果表明该格式对多种流动结构具有良好的捕捉能力,并且同时具有良好的计算效率,去除冗余计算后又降低了约20%的计算时间.

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially nonoscillatory(WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifthorder WENO-JS(WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable.The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On nonuniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime,the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.

An Application of Finite Volume WENO Schemeto Numerical Modeling of Tidal Current

A depth-averaged 2-D numerical model for unsteady tidal flow in estuaries is established by use of the finite volume WENO scheme which maintains both uniform high order accuracy and an essentially non-oscillatory shock transition on unstructured triangular grid. The third order TVD Range-Kutta method is used for time discretization. The model has been firstly tested against four cases： 1） tidal forcing, 2） seiche oscillation, 3） wind setup in a closed bay, and 4） onedimensional dam-break water flow. The results obtained in the present study compare well with those obtained from the corresponding analytic solutions idealized for the above four cases. The model is then applied to the simulation of tidal circulation in the Yangpu Bay, and detailed model calibration and verification have been conducted with measured tidal current in the spring tide, middle tide, and neap tide. The overall performance of the model is in qualitative agreement with the data observed in 2005, and it can be used to calculate the flow in estuaries and coastal waters.

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

A numerical study for WENO scheme-based on different lattice Boltzmann flux solver for compressible flows

In this paper,the finite difference weighted essentially non-oscillatory (WENO) scheme is incorporated into the recently developed four kinds of lattice Boltzmann flux solver (LBFS) to simulate compressible flows,including inviscid LBFS Ⅰ,viscous LBFS Ⅱ,hybrid LBFS Ⅲ and hybrid LBFS Ⅳ.Hybrid LBFS can automatically realize the switch between inviscid LBFS Ⅰ and viscous LBFS Ⅱ through introducing a switch function.The resultant hybrid WENO-LBFS scheme absorbs the advantages of WENO scheme and hybrid LBFS.We investigate the performance of WENO scheme based on four kinds of LBFS systematically.Numerical results indicate that the devopled hybrid WENO-LBFS scheme has high accuracy,high resolution and no oscillations.It can not only accurately calculate smooth solutions,but also can effectively capture contact discontinuities and strong shock waves.

A 3rd Order WENO GLM-MHD Scheme for Magnetic Reconnection

A new numerical scheme of 3rd order Weighted Essentially Non-Oscillatory (WENO) type for 2.5D mixed GLM-MHD in Cartesian coordinates is proposed. The MHD equations are modified by combining the arguments as by Dellar and Dedner et al to couple the divergence constraint with the evolution equations using a Generalized Lagrange Multiplier (GLM). Moreover, the magnetohydrodynamic part of the GLM-MHD system is still in conservation form. Meanwhile, this method is very easy to add to an existing code since the underlying MHD solver does not have to be modified. To show the validation and capacity of its application to MHD problem modelling, interaction between a magnetosonic shock and a denser cloud and magnetic reconnection problems are used to verify this new MHD code. The numerical tests for 2D Orszag and Tang's MHD vortex, interaction between a magnetosonic shock and a denser cloud and magnetic reconnection problems show that the third order WENO MHD solvers are robust and yield reliable results by the new mixed GLM or the mixed EGLM correction here even if it can not be shown that how the divergence errors are transported as well as damped as done for one dimensional ideal MHD by Dedner et al.

WENO格式的一种新型光滑因子及其应用

光滑因子是构造WENO格式的关键,决定了WENO格式能否达到最优收敛精度以及在间断附近能否保持本质无振荡特性。针对广泛应用的五阶WENO格式,采用算子函数近似导数关系的方法,设计了一种新型光滑因子。在间断区域,与传统的光滑因子相比,新型光滑因子对间断的识别更准确。在光滑区域,新型光滑因子使得新型WENO格式的非线性权更接近线性权。理论分析和数值验证表明新型WENO格式即使在一阶临界点也能保持一致五阶精度。一维典型激波问题的数值结果表明,与现有WENO格式相比,新型WENO格式提高了短波的分辨率,降低了格式的耗散,同时能够准确识别间断。对于典型包含激波和剪切层的流动问题,新型WENO格式不仅能够准确地捕捉强激波,而且能够精细模拟剪切层和声波等流场结构。