A Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws

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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Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids

Journal of Computational Physics ◽

10.1016/j.jcp.2013.08.008 ◽

2013 ◽

Vol 255 ◽

pp. 205-227 ◽

Cited By ~ 24

Author(s):

Lucian Ivan ◽

Hans De Sterck ◽

Scott A. Northrup ◽

Clinton P.T. Groth

Keyword(s):

Conservation Laws ◽

Finite Volume ◽

Hyperbolic Conservation Laws ◽

Three Dimensional ◽

Finite Volume Scheme ◽

Dimensional Solution ◽

Hyperbolic Conservation ◽

Volume Scheme ◽

Cubed Sphere

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A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

Communications in Computational Physics ◽

10.4208/cicp.081214.250515s ◽

2015 ◽

Vol 18 (4) ◽

pp. 901-930 ◽

Cited By ~ 9

Author(s):

Ziyao Sun ◽

Honghui Teng ◽

Feng Xiao

Keyword(s):

Conservation Laws ◽

Finite Volume ◽

Solution Structure ◽

Hyperbolic Conservation Laws ◽

High Order ◽

Weno Reconstruction ◽

Hyperbolic Conservation ◽

Average Value ◽

High Order Schemes ◽

Volume Method

AbstractThis paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

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