Improved third‐order weighted essentially nonoscillatory scheme

A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.     

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

In this article, we present two improved third‐order weighted essentially nonoscillatory (WENO) schemes for recovering their design‐order near first‐order critical points. The schemes are constructed in the framework of third‐order WENO‐Z scheme. Two new global smoothness indicators, τ_L3 and τ_L4, are devised by a nonlinear combination of local smoothness indicators (IS_k) and reference values (IS_G) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one‐dimensional linear advection equation or one‐ and two‐dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3‐JS and subsequent WENO3‐N scheme.     

Improvement of the weighted essentially nonoscillatory scheme based on the interaction of smoothness indicators

The weighted essentially nonoscillatory scheme is improved by introducing new smoothness indicators that evaluate the interactions among the classical smoothness indicators suggested by Jiang and Shu. The effect of the key parameters in the new smoothness indicators on the scheme is systematically investigated. The improved scheme has smaller dissipation with larger weight assignment to the discontinuous stencils and higher numerical accuracy with weights closer to the ideal weights. To verify the theory, benchmark problems governed by the linear transport equation, the 1‐dimensional nonlinear Burgers equation, and the Euler equations are conducted and analyzed, respectively. Better computational performances both on numerical resolution and accuracy are shown in the comparisons with other classical weighted essentially nonoscillatory schemes.     

An improved third-order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L_∞ and L₂ norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes.     

A low-dissipation third-order weighted essentially nonoscillatory scheme with a new reference smoothness indicator

The classical third-order weighted essentially nonoscillatory (WENO) scheme is notoriously dissipative as it loses the optimal order of accuracy at critical points and its two-point finite difference in the smoothness indicators is unable to differentiate the critical point from the discontinuity. In recent years, modifications to the smoothness indicators and weights of the classical third-order WENO scheme have been reported to reduce numerical dissipation. This article presents a new reference smoothness indicator for constructing a low-dissipation third-order WENO scheme. The new reference smoothness indicator is a nonlinear combination of the local and global stencil smoothness indicators. The resulting WENO-Rp3 scheme with the power parameter p=1.5 achieves third-order accuracy in smooth regions including critical points and has low dissipation, but numerical results show this scheme cannot keep the ENO property near discontinuities. The recommended WENO-R3 scheme (p=1) keeps the ENO property and performs better than several recently developed third-order WENO schemes.     

Third‐order WENO scheme with a new smoothness indicator

In this article, we have devised a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory (WENO) scheme to achieve desired order of convergence at critical points. In the context of the weighted essentially non‐oscillatory scheme, reference smoothness indicator is constructed in such a way that it satisfies the sufficient condition on the weights for the third‐order convergence. The goal is to construct a reference smoothness indicator such that the resulted scheme have to achieve the required order of accuracy even if the first two derivatives vanish but not the third derivative. The construction of such reference smoothness indicator is not possible through a linear combination of local smoothness indicators only. We have proposed a reference smoothness indicator to be of the fourth order of accuracy on three‐point stencil that contains the linear combination of the first derivative information of the local and global stencils. The performance enhancement of the WENO scheme through this reference smoothness indicator is verified through the standard numerical experiments. Numerical results indicate that the new scheme provides better results in comparison with the earlier third‐order WENO schemes like WENO‐JS and WENO‐Z. Copyright © 2017 John Wiley & Sons, Ltd.     

Novel local smoothness indicators for improving the third‐order WENO scheme

The local smoothness indicators play an important role in the performance of a weighted essentially nonoscillatory (WENO) scheme. Due to having only 2 points available on each substencil, the local smoothness indicators calculated by conventional methods make the third‐order WENO scheme too dissipative. In this paper, we propose a different method to calculate the indicators by using all the 3 points on the global stencil of the third‐order WENO scheme. The numerical results demonstrate that the WENO scheme with the new indicators has less dissipation and better resolution than the conventional third‐order WENO scheme of Jiang and Shu for both smooth and discontinuous solutions.     

A new smoothness indicator for third‐order WENO scheme

We present a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory scheme to recover its design‐order convergence at critical points. This reference smoothness indicator, which involves both the candidate and global smoothness indicators in the weighted essentially non‐oscillatory framework, is devised according to a sufficient condition on the weights for third‐order convergence. The recovery of design‐order is verified by standard tests. Meanwhile, numerical results demonstrate that the present reference smoothness indicator produces sharper representation of the discontinuity owing to the combined effects of larger weight assignment to the discontinuous stencils and convergence rate recovery. Copyright © 2015 John Wiley & Sons, Ltd.     

Improvement of the WENO scheme smoothness estimator

This paper has analyzed the weights of the fifth‐order weighted essentially nonoscillatory scheme suggested by Jiang and Shu and a modified smoothness estimator is suggested to improve the accuracy. Using a function of the minimum and maximum of the original smoothness estimators, the new smoothness estimators have smaller variation among them than the original ones in smooth regions. Hence, as the new weights are closer to the optimal weights, the numerical accuracy is improved. Several numerical experiments are presented to demonstrate the accuracy and robustness of the new scheme. Copyright © 2009 John Wiley & Sons, Ltd.     

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Hermite weighted essentially non‐oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high‐order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193 :115). For example, classical fifth‐order weighted essentially non‐oscillatory (WENO) reconstructions are based on a five‐cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three‐cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth‐order accurate well‐balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five‐step and fourth‐order accurate method. Besides the classical SWE, the non‐homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well‐balanced treatment of the source term involved in such equations is developed and tested. Several standard one‐dimensional test cases are used to verify the high‐order accuracy, the C‐property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.     

A well‐balanced weighted essentially non‐oscillatory scheme for pollutant transport in shallow water

In this paper, a well‐balanced finite difference weighted essentially non‐oscillatory scheme is presented for modeling transport and diffusion of pollutant in shallow water flows. The scheme balances exactly the flux gradients and the source terms. Extensive one‐dimensional and two‐dimensional numerical experiments on uniform and curvilinear meshes strongly suggest that high resolution results are achieved for both water depth and pollutant concentration. The scheme is efficient and robust and can be applied to practical numerical simulation of pollutant transport phenomena in shallow water flows. Copyright © 2012 John Wiley & Sons, Ltd.     

无粘可压缩流动的改进高精度方法

为更准确捕捉复杂流场的流动细节,通过对WENO格式的光滑因子进行改进,发展了一种新的五阶WENO格式。对三阶ENO格式进行加权可以得到五阶WENO格式,但是不同的加权处理,WENO格式在极值处保持加权基本无振荡的效果不同,本文构造了二阶精度的局部光滑因子,及不含一阶二阶导数的高阶全局光滑因子,从而实现WENO格式在极值处有五阶精度。基于改进五阶WENO格式,对一维对流方程、一维和二维可压缩无粘问题进行算例验证,并与传统WENO-JS格式和WENO-Z格式进行比较。计算结果表明,改进五阶WENO格式有较高的精度和收敛速度,有较低的数值耗散,能有效捕捉间断、激波和涡等复杂流动。     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.     

一种提高极值点处收敛精度的三阶WENO-Z格式

为了提高三阶WENO-Z格式在极值点处的计算精度,通过理论推导给出三阶WENO格式满足收敛精度的充分条件。采用泰勒级数展开的方式,推导给出所构造格式非线性权重的计算公式,并综合权衡计算精度和计算稳定性确定所构造格式的参数。通过两个典型的精度测试,验证了改进格式在光滑流场极值点区域逼近三阶精度。进一步选用激波与熵波相互作用和Richtmyer-Meshkov不稳定性等经典算例,证实了本文提出的改进格式WENO-PZ3相较其他格式（WENO-JS3和WENO-Z3）不仅具有较高的精度,而且降低了格式的耗散,提高了对流场结构的分辨率。     

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

New numerical schemes based on a criterion for constructing essentially stable and accurate numerical schemes for convection-dominated equations

In order to obtain stable and accurate numerical solutions for the convection-dominated steady transport equations, we propose a criterion for constructing numerical schemes for the convection term that the roots of the characteristic equation of the resulting difference equation have poles. By imposing this criterion on the difference coefficients of the convection term, we construct two numerical schemes for the convection-dominated equations. One is based on polynomial differencing and the other on locally exact differencing. The former scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $, which is the critical value for its stability, while it approaches the second-order upwind scheme as Rm goes to infinity. Hence the former scheme interpolates a stable scheme between the QUICK scheme at Rm = $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $ and the second-order upwind scheme at Rm = ∞. Numerical solutions with the present new schemes for the one-dimensional, linear, steady convection-diffusion equations showed good results.     

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.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient three-level explicit time-split scheme for solving two-dimensional unsteady nonlinear coupled Burgers' equations

This work deals with the numerical solutions of two-dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three-level explicit time-split MacCormack approach. In this technique, the differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second-order accurate in time and fourth-order convergent in space, whereas it is second-order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two-dimensional time-dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed.     

Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction

High-order finite volume schemes for conservation laws are very useful in applications, due to their ability to compute accurate solutions on quite coarse meshes and with very few restrictions on the kind of cells employed in the discretization. For balance laws, the ability to approximate up to machine precision relevant steady states allows the scheme to compute accurately, also on coarse meshes, small perturbations of such states, which are very relevant for many applications. In this paper, we propose third- and fourth-order accurate finite volume schemes for the shallow water equations. The schemes have the well-balanced property thanks to a path-conservative approach applied to an appropriate nonconservative reformulation of the equations. High-order accuracy is achieved by designing truly two-dimensional (2D) reconstruction procedures of the central WENO (CWENO ) type. The novel schemes are tested for accuracy and well-balancing and shown to maintain positivity of the water height on wet/dry transitions. Finally, they are applied to simulate the Tohoku 2011 tsunami event.