An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws (original) (raw)

Optimized Weighted Essentially Nonoscillatory Third-Order Schemes for Hyperbolic Conservation Laws

Journal of Applied Mathematics, 2013

We describe briefly how a third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling a WENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger’s equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lowerL1errors. Lastly, we test numerically the order of convergence of the WENO3 scheme.

High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws

Journal of Computational Physics, 2011

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.

A Modified Fifth Order Finite Difference Hermite WENO Scheme for Hyperbolic Conservation Laws

Journal of Scientific Computing

In this paper, we develop a modified fifth order accuracy finite difference Hermite WENO (HWENO) scheme for solving hyperbolic conservation laws. The main idea is that we first modify the derivatives of the solution by Hermite WENO interpolations, then we discretize the original and derivative equations in the spatial directions by the same approximation polynomials. Comparing with the original finite difference HWENO scheme of Liu and Qiu (J Sci Comput 63:548-572, 2015), one of the advantages is that the modified HWENO scheme is more robust than the original one since we do not need to use the additional positivity-preserving flux limiter methodology, and larger CFL number can be applied. Another advantage is that higher order numerical accuracy than the original scheme can be achieved for two-dimensional problems under the condition of using the same approximation stencil and information. Furthermore, the modified scheme preserves the nice property of compactness shared by HWENO schemes, i.e., only immediate neighbor information is needed in the reconstruction, and it has smaller numerical errors and higher resolution than the classical fifth order finite difference WENO scheme of Jiang and Shu (J Comput Phys 126:202-228, 1996). Various benchmark numerical tests of both one-dimensional and twodimensional problems are presented to illustrate the numerical accuracy, high resolution and robustness of the proposed novel HWENO scheme.

A new weighted essentially non-oscillatory WENO-NIP scheme for hyperbolic conservation laws

Computers & Fluids, 2019

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Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws

Journal of Computational Physics, 2007

In this article we introduce the multi-domain hybrid Spectral-WENO method aimed at the discontinuous solutions of hyperbolic conservation laws. The main idea is to conjugate the non-oscillatory properties of the high order weighted essentially non-oscillatory (WENO) finite difference schemes with the high computational efficiency and accuracy of spectral methods. Built in a multi-domain framework, subdomain adaptivity in space and time is used in order to maintain the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains while the smooth parts of the solution are kept in spectral ones. A high order multi-resolution algorithm by Ami Harten [6]is used to determine the smoothness of the solution in each subdomain. Numerical experiments with the simulation of compressible flow in the presence of shock waves are performed.

A Hybrid WENO Scheme for Conservation Laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A fourth-order total variation diminishing (TVD) finite difference scheme is conjugated with the seven-order WENO scheme. An efficient multi-resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with the seven-order WENO scheme while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one-and two-dimensional problems are presented.

A Robust TVD-WENO Scheme for Conservation Laws

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 2011

The ultimate goal of this article is to develop a robust and accurate numerical method for solving hyperbolic conservation laws in one and two dimensions. A hybrid numerical method, coupling a cheap fourth order total variation diminishing (TVD) scheme [1] for smooth region and a Robust seventh-order weighted non-oscillatory (WENO) scheme [2] near discontinuities, is considered. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme, while the smooth regions are computed with fourth order total variation diminishing (TVD). 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.

A Fifth Order Alternative Mapped WENO Scheme for Nonlinear Hyperbolic Conservation Laws

2021

In this work, we have developed a fifth-order alternative mapped weighted essentially nonoscillatory (AWENO-M) finite volume scheme using non-linear weights of mapped WENO reconstruction scheme of Henrick et al. (J. Comput. Phys., 207 (2005), pp. 542–567) for solving hyperbolic conservation laws. The reconstruction of numerical flux is done using primitive variables instead of conservative variables. The present scheme results in less spurious oscillations near discontinuities and shows higher-order accuracy at critical points compared to the alternative WENO scheme (AWENO) based on traditional non-linear weights of Jiang and Shu (J. Comput. Phys., 228 (1996), pp. 202–228). The third-order Runge-Kutta method has been used for solution advancement in time. The Harten-Lax-van Leer-Contact (HLLC) shock-capturing method is used to provide necessary upwinding into the solution. The performance of the present scheme is evaluated in terms of accuracy, computational cost, and resolution of ...

A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws

2008

We propose a new fourth-order non-oscillatory central scheme for computing approximate solutions of hyperbolic conservation laws. A piecewise cubic polynomial is used for the spatial reconstruction and for the numerical derivatives we choose genuinely fourth-order accurate non-oscillatory approximations. The solution is advanced in time using natural continuous extension of Runge-Kutta methods. Numerical tests on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and sharper than existing fourth-order central schemes when solving profiles with discontinuities. Experiments on nonlinear Burgers' equation indicate that our scheme is superior to existing fourth-order central schemes in the sense that the total variation of the computed solutions are closer to the total variation of the exact solution.

Verification of WENO for hyperbolic conservation laws

Anais do ... Congresso Ibero-Latino-Americano de Métodos Computacionais em Engenharia, 2017

Verification is an important procedure to assess numerical schemes and its solutions. One of the most popular numerical scheme is the Weighted Essentially Nonoscillatory (WENO). This scheme is high-order accurate and presents high resolution. The purpose of this work is to assess numerical solutions of three hyperbolic equations, namely, the linear advection equation and one-dimensional and two-dimensional Euler system of equations. In order to solve these equations, the Finite Volume Method was employed with an explicit formulation, Lax-Friedrichs flux, two numerical shcemes, first-order and WENO, and Runge-Kutta method. We have presented an error and order analysis for both numerical schemes, in which the WENO scheme showed fifth-order accuracy in all problems, except with discontinuous solutions as expected. We also confirmed that the WENO scheme is less dissipative even with discontinuities and in coarse meshes.