A fifth-order nonlinear spectral difference scheme for hyperbolic conservation laws

2021 ◽  
Vol 221 ◽  
pp. 104928
Author(s):  
Yu Lin ◽  
Yaming Chen ◽  
Xiaogang Deng
Keyword(s):  
Difference Scheme ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Spectral Difference ◽  
Hyperbolic Conservation ◽  
Fifth Order

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.

Fully and Semi-discrete Fourth-order Schemes for Hyperbolic Conservation Laws

2007 ◽  
Vol 7 (3) ◽  
pp. 264-282
Author(s):  
Y.H. Zahran
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Hyperbolic Conservation Laws ◽  
Tvd Scheme ◽  
Test Problems ◽  
Hyperbolic Conservation ◽  
Runge Kutta Method ◽  
Order Scheme ◽  
Spurious Oscillations ◽  
Fifth Order

AbstractA new fourth order accurate centered finite difference scheme for the solution of hyperbolic conservation laws is presented. A technique of making the fourth order scheme TVD is presented. The resulting scheme can avoid spurious oscillations and preserve fourth order accuracy in smooth parts. We discuss the extension of the TVD scheme to the nonlinear scalar hyperbolic conservation laws. For nonlinear systems, the TVD constraint is applied by solving shallow water equations. Then, we propose to use this fourth order flux as a building block in spatially fifth order weighted essentially non-oscillatory (WENO) schemes. The numerical solution is advanced in time by the third order TVD Runge — Kutta method. The performance of the scheme is assessed by solving test problems. The numerical results are presented and compared to the exact solutions and other methods.

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