A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36], [33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.     

An adaptive mesh method for 1D hyperbolic conservation laws

An adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws, which combines the rezoning approach with the finite volume weighted essentially non-oscillatory (WENO) scheme. An a posteriori error estimate, used to equidistribute the mesh, is obtained from the differences between respective numerical solutions of 5th-order WENO (WENO5) and 3rd-order ENO (ENO3) schemes. The number of grids can be adaptively readjusted based on the solution structure. For higher efficiency, mesh readjustment is performed every few time steps rather than every time step. In addition, a high order conservative interpolation is used to compute the physical solutions on the new mesh from old mesh based on the finite volume ENO reconstruction. Extensive examples suggest that this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.     

A new fifth order finite difference WENO scheme for Hamilton‐Jacobi equations

In this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non‐oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton‐Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1095–1113, 2017     

RBF‐ENO/WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations

In this research, a class of radial basis functions (RBFs) ENO/WENO schemes with a Lax–Wendroff time discretization procedure, named as RENO/RWENO‐LW, for solving Hamilton–Jacobi (H–J) equations is designed. Particularly the multi‐quadratic RBFs are used. These schemes enhance the local accuracy and convergence by locally optimizing the shape parameters. Comparing with the original WENO with Lax–Wendroff time discretization schemes of Qiu for HJ equations, the new schemes provide more accurate reconstructions and sharper solution profiles near strong discontinuous derivative. Also, the RENO/RWENO‐LW schemes are easy to implement in the existing original ENO/WENO code. Extensive numerical experiments are considered to verify the capability of the new schemes.     

An efficient and conservative hybrid method for solving multidimensional conservation laws

We have developed an efficient hybrid technique for solving nonlinear conservation laws. Previous hybrid techniques have been accurate but lacked the property of conservation, whereas our technique is both accurate and conservative. To achieve this, we superimposed all possible stencils for ENO polynomials and weighted the value from each cells in a way that depends on the numerical solution. Computational efficiency relies on switching from central data where the exact solution is smooth to noncentral data near discontinuities. We prove the theoretical consistency of the technique and discuss the connection with ENO and WENO methods. We introduce time dependency by combining our method with Runge‐Kutta schemes that are TVD preserving. We have verified our technique experimentally by solving a suite of test problems with convex and non‐convex flux functions taken from the literature. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

基于ENO格式的三阶修正系数格式

不增加基点,仅摄动二阶ENO格式的系数(简记为MCENO),得到一类求解双曲型守恒律方程的三阶MCENO格式．由MCENO格式的构造过程可以看出,MCENO格式保留了ENO格式的许多性质,例如本质无振荡性、TVB性质等,且能提高一阶精度．进一步,利用MCENO格式模拟二维Rayleigh-Taylor(RT)不稳定性和Lax激波管的数值求解问题．数值结果表明,t=2.0时,MCENO格式的密度曲线处于三阶WENO格式和五阶WENO格式之间,是一个高效高精度格式．值得注意的是,三阶MCENO格式,三阶WENO格式和五阶WENO格式的CPU时间之比为0.62:1:2.19．表明相对于原始ENO格式,MCENO格式在光滑区域有较高精度,能提高格式精度．     

An efficient TVD‐WENO method for conservation laws

In this article, we present a high‐resolution hybrid scheme for solving hyperbolic conservation laws in one and two dimensions. In this scheme, we use a cheap fourth order total variation diminishing (TVD) scheme for smooth region and expensive seventh order weighted nonoscillatory (WENO) scheme near discontinuities. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multiresolution technique. For time integration, we use the third order TVD Runge‐Kutta scheme. The accuracy of the resulting hybrid high order scheme is comparable with these of WENO, but with significant decrease of the CPU cost. Numerical demonstrates that the proposed scheme is comparable to the high order WENO scheme and superior to the fourth order TVD scheme. Our scheme has the added advantage of simplicity and computational efficiency. Numerical tests are presented which show the robustness and effectiveness of the proposed scheme.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical experiments on a hybrid WENO5 filter for shock‐capturing

In the present paper, a hybrid filter is introduced for high accurate numerical simulation of shock‐containing flows. The fourth‐order compact finite difference scheme is used for the spatial discretization and the third‐order Runge–Kutta scheme is used for the time integration. After each time‐step, the hybrid filter is applied on the results. The filter is composed of a linear sixth‐order filter and the dissipative part of a fifth‐order weighted essentially nonoscillatory scheme (WENO5). The classic WENO5 scheme and the WENO5 scheme with adaptive order (WENO5‐AO) are used to form the hybrid filter. Using a shock‐detecting sensor, the hybrid filter reduces to the linear sixth‐order filter in smooth regions for damping high frequency waves and reduces to the WENO5 filter at shocks in order to eliminate unwanted oscillations produced by the nondissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the advection, Euler and Navier–Stokes equations. The results are compared with that of a hybrid second‐order filter and also that of the WENO5 and WENO5‐AO schemes.     

ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT

In this paper we further explore and apply our recent anti-diffusive flux corrected highorder finite difference WENO schemes for conservation laws [18] to compute the Saint-Venant system of shallow water equations with pollutant propagation, which is describedby a transport equation. The motivation is that the high order anti-diffusive WENOscheme for conservation laws produces sharp resolution of contact discontinuities whilekeeping high order accuracy for the approximation in the smooth region of the solution.The application of the anti-diffusive high order WENO scheme to the Saint-Venant systemof shallow water equations with transport of pollutant achieves high resolution     

ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES Ⅱ:CELL AVERAGES AND MULTIRESOLUTION

This paper is the second part of the article and is devoted to the construction and analysis of new non-linear optimal weights for WENO interpolation capable of rising the order of accuracy close to discontinuities for data discretized in the cell averages. Thus, now we are interested in analyzing the capabilities of the new algorithm when working with functions belonging to the subspace $L^1\cap L^2$ and that, consequently, are piecewise smooth and can present jump discontinuities. The new non-linear optimal weights are redesigned in a way that leads to optimal theoretical accuracy close to the discontinuities and at smooth zones. We will present the new algorithm for the approximation case and we will analyze its accuracy. Then we will explain how to use the new algorithm in multiresolution applications for univariate and bivariate functions. The numerical results confirm the theoretical proofs presented.     

Mapped WENO schemes based on a new smoothness indicator for Hamilton–Jacobi equations

In this paper, we introduce an improved version of mapped weighted essentially non-oscillatory (WENO) schemes for solving Hamilton–Jacobi equations. To this end, we first discuss new smoothness indicators for WENO construction. Then the new smoothness indicators are combined with the mapping function developed by Henrick et al. (2005) [31]. The proposed scheme yields fifth-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. Numerical experiments are provided to demonstrate the performance of the proposed schemes on a variety of one-dimensional and two-dimensional problems.     

Numerical Study of Oscillatory Solutions of the Gas-Dynamic Equations

Solutions of the one-dimensional gas-dynamic equations with oscillatory initial data are studied numerically. A fourth-order ENO scheme is used, together with an artificial compression method to effectively eliminate the smearing of contact discontinuities. The numerical results show that the oscillations die out in velocity and pressure, and persist in density. A system of equations governing the propagation of these oscillations is derived.     

High-order accurate dissipative weighted compact nonlinear schemes

Based on the method deriving dissipative compact linear schemes (DCS), novel high-order dissipative weighted compact nonlinear schemes (DWCNS) are developed. By Fourier analysis,the dissipative and dispersive features of DWCNS are discussed. In view of the modified wave number, the DWCNS are equivalent to the fifth-order upwind biased explicit schemes in smooth regions and the interpolations at cell-edges dominate the accuracy of DWCNS. Boundary and near boundary schemes are developed and the asymptotic stabilities of DWCNS on both uniform and stretching grids are analyzed. The multi-dimensional implementations for Euler and Navier-Stokes equations are discussed. Several numerical inviscid and viscous results are given which show the good performances of the DWCNS for discontinuities capturing, high accuracy for boundary layer resolutions, good convergent rates (the root-mean-square of residuals approaching machine zero for solutions with strong shocks) and especially the damping effect on the spudous oscillations which were found in the solutions obtained by TVD and ENO schemes.     

Edge detection from truncated Fourier data using spectral mollifiers

Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the “sharp peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise.     

Local edge detectors using a sigmoidal transformation for piecewise smooth data

For piecewise smooth data, edges can be recognized by jump discontinuities in the data. Successful edge detection is essential in digital signal processing as the most relevant information is often observed near the edges in each segmented region. In this paper, using the concentration property of existing local edge detectors and the clustering property of sigmoidal transformations, we provide enhanced edge detectors which diminish the oscillations of the local detector near jump discontinuities as well as highly improve rate of convergence away from the discontinuities. Numerical results of some examples illustrate efficiency of the presented method.     

Binary weighted essentially non-oscillatory (BWENO) approximation

Weighted essentially non-oscillatory (WENO) schemes have been mainly used for solving hyperbolic partial differential equations (PDEs). Such schemes are capable of high order approximation in smooth regions and non-oscillatory sharp resolution of discontinuities. The base of the WENO schemes is a non-oscillatory WENO approximation procedure, which is not necessarily related to PDEs. The typical WENO procedures are WENO interpolation and WENO reconstruction. The WENO algorithm has gained much popularity but the basic idea of approximation did not change much over the years. In this paper, we first briefly review the idea of WENO interpolation and propose a modification of the basic algorithm. New approximation should improve basic characteristics of the approximation and provide a more flexible framework for future applications. New WENO procedure involves a binary tree weighted construction that is based on key ideas of WENO algorithm and we refer to it as the binary weighted essentially non-oscillatory (BWENO) approximation. New algorithm comes in a rational and a polynomial version. Furthermore, we describe the WENO reconstruction procedure, which is usually involved in the numerical schemes for hyperbolic PDEs, and propose the new reconstruction procedure based on the described BWENO interpolation. The obtained numerical results show that the newly proposed procedures perform very well on the considered test examples.     

A HIGHLY ACCURATE NUMERICAL METHOD FOR FLOW PROBLEMS WITH INTERACTIONS OF DISCONTINUITIES

1. IntroductionCompared with the widely used shock capturing methods for the compressible flows, theshock fitting methods have the main advantage of accuracy. The shock fitting methods areusually very accurate wherever it can be applied. Several kinds of shock fitting methods havebeen developed il1 the last tl1ree decades. Glimrn and his coworkers have worked extensively onthe front tracking methods and applied them to many complicated problems with shocks andother types of singularities [1]-[…     

Symmetrical weighted essentially non‐oscillatory‐flux limiter schemes for Hamilton–Jacobi equations

In this paper, we propose a new scheme that combines weighted essentially non‐oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton–Jacobi equations. In one‐dimensional (1D) case, first, we obtain an optimum polynomial on a four‐point stencil. This optimum polynomial is third‐order accurate in regions of smoothness. Next, we modify a second‐order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction‐evolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the non‐oscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

基于新光滑因子的WENO5格式

针对WENO格式的构造,本文给出了一个WENO为5阶的充分条件,降低了Henrick等人提出的充分条件对于权因子的精度要求.另外,对于Jiang和Shu提出的WENO5中的光滑因子中两项的系数做出了调整,并结合Borges等人的方法得到了新的WENO权因子计算方法.从数值试验的结果可以看出,新的WENO格式对于连续波形...