scholarly journals Third order TVD scheme for hyperbolic conservation laws

2007 ◽  
Vol 14 (2) ◽  
pp. 259-275 ◽  
Author(s):  
Yousef Hashem Zahran
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Tvd Scheme ◽  
Third Order ◽  
Hyperbolic Conservation

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.

AN EFFICIENT THIRD-ORDER SCHEME FOR THREE-DIMENSIONAL HYPERBOLIC CONSERVATION LAWS

2012 ◽  
Vol 03 (04) ◽  
pp. 1250015
Author(s):  
LI CAI ◽  
JIAN-HU FENG ◽  
YU-FENG NIE ◽  
WEN-XIAN XIE
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Three Dimensional ◽  
Upwind Scheme ◽  
Third Order ◽  
Hyperbolic Conservation ◽  
Weighted Essentially Nonoscillatory ◽  
Order Scheme ◽  
Cweno Reconstruction ◽  
Three Space

In this paper, we present a third-order central weighted essentially nonoscillatory (CWENO) reconstruction for computations of hyperbolic conservation laws in three space dimensions. Simultaneously, as a Godunov-type central scheme, the CWENO-type central-upwind scheme, i.e., the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction, is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies. The high resolution, the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.

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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