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.

scholarly journals A Weno-Tvd Implementation for Solving Some Problems of Hyperbolic Conservation Laws

2018 ◽  
Author(s):  
Jhon Alberto Polo Vásquez ◽  
Miguel Antonio Caro Candezano
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Computer Code ◽  
Tvd Scheme ◽  
Hyperbolic Conservation ◽  
Flux Limiter ◽  
Systems Of Conservation Laws ◽  
Fifth Order

This work deals with a numerical implementation of a fifth order CENTRAL WENO-TVD (\textit{Weighted Essentially Non-Oscillatory-Total Variation Dimimishing}) of Haschem (2006) scheme applied to the convective terms of some hyperbolic conservation laws problems, in a volume finite framework. The WENO-TVD scheme is used to solve the 1D advection and Burgers equations. For this case is implemented two different numerical fluxes: The Lax-Friedrichs and TVD fluxes. In the TVD fluxes the schemes applied are in flux-limiter form. The schemes implemented for this flux are: Van Albada-1 (van Albada et al.,1982), van Albada-2 (Kermani et al., 2003), van Leer (Hassanzadeh, 2009) and MINMOD (Hirsch, 2007). The WENO type schemes are characterized for their high order approximation, and do not produce spurious oscilations near discontinuities, shocks and higher gradients. A third order Runge-Kutta TVD for the temporal variable is used. Qualitative and quantitative comparison are presented. The numerical solutions are computed with an in-house computer code developed in MATLAB software. In future works, it will develope a paralelization of computer code for solving systems of conservation laws, e.g. Euler equations of gas dynamics.

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.

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