scholarly journals A fourth-order entropy condition scheme for systems of hyperbolic conservation laws

2021 ◽  
Vol 15 (1) ◽  
pp. 1259-1281
Author(s):  
Tong Zhou ◽  
Haitao Dong
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Hyperbolic Conservation Laws ◽  
Entropy Condition ◽  
Hyperbolic Conservation

scholarly journals A Fourth Order Entropy Stable Scheme for Hyperbolic Conservation Laws

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 508 ◽  
Author(s):  
Xiaohan Cheng
Keyword(s):  
Conservation Laws ◽  
Fourth Order ◽  
Hyperbolic Conservation Laws ◽  
Order Accuracy ◽  
Hyperbolic Conservation ◽  
Diffusion Operator ◽  
Sign Property ◽  
Entropy Stable ◽  
Order Four ◽  
Stable Scheme

This paper develops a fourth order entropy stable scheme to approximate the entropy solution of one-dimensional hyperbolic conservation laws. The scheme is constructed by employing a high order entropy conservative flux of order four in conjunction with a suitable numerical diffusion operator that based on a fourth order non-oscillatory reconstruction which satisfies the sign property. The constructed scheme possesses two features: (1) it achieves fourth order accuracy in the smooth area while keeping high resolution with sharp discontinuity transitions in the nonsmooth area; (2) it is entropy stable. Some typical numerical experiments are performed to illustrate the capability of the new entropy stable scheme.

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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