A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws

2021 ◽  
Vol 13 (5) ◽  
pp. 1064-1095
Author(s):  
global sci
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Low Dissipation

scholarly journals Towards Building the OP-Mapped WENO Schemes: A General Methodology

2021 ◽  
Vol 26 (4) ◽  
pp. 67
Author(s):  
Ruo Li ◽  
Wei Zhong
Keyword(s):  
Conservation Laws ◽  
Numerical Experiments ◽  
Hyperbolic Conservation Laws ◽  
Critical Role ◽  
Weno Scheme ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Low Dissipation ◽  
The Family ◽  
Spurious Oscillations

A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions, but in the meantime prevent spurious oscillations in the solving of hyperbolic conservation laws with long output times. Our goal for this article was to address this widely known problem. In our previous work, the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long simulations, and hence it was indicated that the OP criterion plays a critical role in the maintenance of low-dissipation and robustness for mapped WENO schemes. Thus, in our present work, we firstly defined the family of mapped WENO schemes, whose mappings meet the OP criterion, as OP-Mapped WENO. Next, we attentively took a closer look at the mappings of various existing mapped WENO schemes and devised a general formula for them. That helped us to extend the OP criterion to the design of improved mappings. Then, we created a generalized implementation of obtaining a group of OP-Mapped WENO schemes, named MOP-WENO-X, as they are developed from the existing mapped WENO-X schemes, where the notation “X” is used to identify the version of the existing mapped WENO scheme. Finally, extensive numerical experiments and comparisons with competing schemes were conducted to demonstrate the enhanced performances of the MOP-WENO-X schemes.

WENO-TVD schemes for hyperbolic conservation laws

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Building Blocks ◽  
Second Order ◽  
Third Order ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Systematic Assessment ◽  
The Third ◽  
Seventh Order

The purpose of this paper is twofold. Firstly we carry out a modification of the finite volume WENO (weighted essentially non-oscillatory) scheme of Titarev and Toro [14] and [15].This modification is done by using two fluxes as building blocks in spatially fifth order WENO schemes instead of the second order TVD flux proposed by Titarev and Toro [14] and [15]. These fluxes are the second order TVD flux [19] and the third order TVD flux [20].Secondly, we propose to use these fluxes as a building block in spatially seventh order WENO schemes. The numerical solution is advanced in time by the third order TVD Runge–Kutta method. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws, in one and two dimension is presented. Systematic assessment of the proposed schemes shows substantial gains in accuracy and better resolution of discontinuities, particularly for problems involving long time evolution containing both smooth and non-smooth features.

A homotopy method based on WENO schemes for solving steady state problems of hyperbolic conservation laws

2013 ◽  
Vol 250 ◽  
pp. 332-346 ◽  
Author(s):  
Wenrui Hao ◽  
Jonathan D. Hauenstein ◽  
Chi-Wang Shu ◽  
Andrew J. Sommese ◽  
Zhiliang Xu ◽  
...  
Keyword(s):  
Steady State ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Homotopy Method ◽  
Weno Schemes ◽  
Hyperbolic Conservation

High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study

2016 ◽  
Vol 20 (4) ◽  
pp. 835-869 ◽  
Author(s):  
Liang Wu ◽  
Yong-Tao Zhang ◽  
Shuhai Zhang ◽  
Chi-Wang Shu
Keyword(s):  
Fixed Point ◽  
Steady State ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Efficient Computation ◽  
Weno Schemes ◽  
Hyperbolic Conservation ◽  
Time Marching ◽  
Fifth Order

AbstractFixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.

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