Compact Third-Order Logarithmic Limiting for Nonlinear Hyperbolic Conservation Laws

2008 ◽  
pp. 347-354 ◽  
Author(s):  
M. Čada ◽  
M. Torrilhon ◽  
R. Jeltsch
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Third Order ◽  
Hyperbolic Conservation ◽  
Nonlinear Hyperbolic Conservation Laws

A third-order entropy stable scheme for hyperbolic conservation laws

2016 ◽  
Vol 13 (01) ◽  
pp. 129-145 ◽  
Author(s):  
Xiaohan Cheng ◽  
Yufeng Nie
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Property A ◽  
Third Order ◽  
Dissipative Term ◽  
Hyperbolic Conservation ◽  
Diffusion Operator ◽  
Entropy Stable ◽  
Nonlinear Hyperbolic Conservation Laws ◽  
Stable Scheme

A third-order entropy stable scheme for nonlinear hyperbolic conservation laws is proposed here. This scheme contains two main ingredients: a fourth-order entropy conservative flux and a third-order numerical diffusion operator. A piecewise-quadratic reconstruction from pointwise values is developed in order to approximate the third-order dissipative term. To guarantee a non-oscillating property, a nonlinear limiter is employed and, furthermore, the scheme is proven to be entropy stable. Finally, numerical experiments are presented and demonstrate the accuracy, high-resolution, and robustness of our method.

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.

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