Journal ArticleDOI
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
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TLDR
The history and basic formulation of WENO schemes are reviewed, the main ideas in using WenO schemes to solve various hyperbolic PDEs and other convection dominated problems are outlined, and a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications are presented.Abstract:
High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.read more
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Linear and nonlinear waves
TL;DR: The study of waves can be traced back to antiquity where philosophers, such as Pythagoras, studied the relation of pitch and length of string in musical instruments and the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt in his treatise Theory of Sound.
Journal ArticleDOI
On maximum-principle-satisfying high order schemes for scalar conservation laws
Xiangxiong Zhang,Chi-Wang Shu +1 more
TL;DR: It is shown that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressibles velocity field.
Journal ArticleDOI
Numerical Methods for High-Speed Flows
TL;DR: In this paper, the authors review numerical methods for direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent compressible flow in the presence of shock waves.
Journal Article
High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations
TL;DR: In this article, central weighted essentially nonoscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.
References
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Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes
TL;DR: In this article, it is shown that these features can be obtained by constructing a matrix with a certain property U, i.e., property U is a property of the solution of the Riemann problem.
Journal ArticleDOI
Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method
TL;DR: In this article, a second-order extension of the Lagrangean method is proposed to integrate the equations of ideal compressible flow, which is based on the integral conservation laws and is dissipative, so that it can be used across shocks.
Journal ArticleDOI
Compact finite difference schemes with spectral-like resolution
TL;DR: In this article, the authors present finite-difference schemes for the evaluation of first-order, second-order and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes.
Book
Riemann Solvers and Numerical Methods for Fluid Dynamics
TL;DR: In this article, the authors present references and index Reference Record created on 2004-09-07, modified on 2016-08-08 and a reference record created on 2003-09 -07.
Journal ArticleDOI
Efficient Implementation of Weighted ENO Schemes
Guang-Shan Jiang,Chi-Wang Shu +1 more
TL;DR: A new way of measuring the smoothness of a numerical solution is proposed, emulating the idea of minimizing the total variation of the approximation, which results in a fifth-order WENO scheme for the caser= 3, instead of the fourth-order with the original smoothness measurement by Liuet al.
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