scholarly journals Optimized Weighted Essentially Nonoscillatory Third-Order Schemes for Hyperbolic Conservation Laws

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
A. R. Appadu ◽  
A. A. I. Peer
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Temporal Integration ◽  
Weno Scheme ◽  
Integration Scheme ◽  
Advection Equation ◽  
Third Order ◽  
Hyperbolic Conservation ◽  
Weighted Essentially Nonoscillatory ◽  
Linear Advection Equation ◽  
Essentially Nonoscillatory

We describe briefly how a third-order Weighted Essentially Nonoscillatory (WENO) scheme is derived by coupling a WENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative properties when used to approximate the 1D linear advection equation and use a technique of optimisation to find the optimal cfl number of the scheme. We carry out some numerical experiments dealing with wave propagation based on the 1D linear advection and 1D Burger’s equation at some different cfl numbers and show that the optimal cfl does indeed cause less dispersion, less dissipation, and lowerL1errors. Lastly, we test numerically the order of convergence of the WENO3 scheme.

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.

scholarly journals Construction of third-order schemes using Lagrange-Burmann expansions for the numerical integration of inviscid gas equations

2016 ◽  
pp. 21-43
Author(s):  
Е.В. Ворожцов
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Numerical Solutions ◽  
Advection Equation ◽  
Gas Flows ◽  
Third Order ◽  
Order Of Accuracy ◽  
Order Difference ◽  
Hyperbolic Conservation ◽  
Inviscid Gas ◽  
The One

Предлагается строить явные разностные схемы третьего порядка точности для гиперболических законов сохранения с применением разложений сеточных функций в ряды Лагранжа--Бюрмана. Результаты тестовых расчетов для случаев одномерного уравнения переноса и многомерных уравнений Эйлера невязкого сжимаемого газа подтверждают третий порядок точности построенных схем. Получены квазимонотонные профили численных решений. It is proposed to construct several explicit third-order difference schemes for the hyperbolic conservation laws using the expansions of grid functions in Lagrange-Burmann series. The results of test computations for the one-dimensional advection equation and multidimensional Euler equations governing the inviscid compressible gas flows confirm the third order of accuracy of the constructed schemes. The quasi-monotonous profiles of numerical solutions are obtained.

scholarly journals Entropy stable essentially nonoscillatory methods based on RBF reconstruction

2019 ◽  
Vol 53 (3) ◽  
pp. 925-958 ◽  
Author(s):  
Jan S. Hesthaven ◽  
Fabian Mönkeberg
Keyword(s):  
Radial Basis Functions ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Basis Functions ◽  
One Dimensional ◽  
Hyperbolic Conservation ◽  
Radial Basis ◽  
Sign Property ◽  
Entropy Stable ◽  
Essentially Nonoscillatory

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

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