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Modifying and Reducing Numerical Dissipation in A Two-Dimensional Central-Upwind Scheme

Published online by Cambridge University Press:  03 June 2015

Chi-Jer Yu  and
Chii-Tung Liu
Chi-Jer Yu*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan
Chii-Tung Liu
Affiliation:
Department of Computer Science and Information Engineering, Chaoyang University of Technology, 168 Jifong E. Rd., Wufong Township Taichung County 41349, Taiwan
*
Corresponding author. URL:http://www.math.nctu.edu.tw/faculty/e_faculty_content.php?S_ID=31&SC_ID=1, Email: ycj@math.nctu.edu.tw
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Abstract

This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation. The prototype, extended from a 1D model, reduces substantially less dissipation than expected. The problem arises from over-restriction of some slope limiters, which keep slopes between interfaces of cells to be Total-Variation-Diminishing. This study reports the defect and presents a re-derived optimal formula. Numerical experiments highlight the significance of this formula, especially in long-time, large-scale simulations.

Keywords

76T0565M05Hyperbolic systems of conservation lawsGodunov-type finite-volume methodscentral-upwind schemeKurganovnumerical dissipationanti-diffusion
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 3 , June 2012 , pp. 340 - 353
Copyright
Copyright © Global-Science Press 2012

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References

[1]Antoniou, A., Karantasis, K., Polychronopoulos, E. and Ekaterinaris, J., Acceleration of a finite-difference WENO scheme for large-scale simulations on many-core architectures, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010, AIAA Paper 2010–0525.Google Scholar
[2]Kurganov, A. and Lin, C.-T., On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys., 2 (2007), pp. 141–163.Google Scholar
[3]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, Comput. Phys., 160 (2000), pp. 241–282.Google Scholar
[4]Kurganov, A. and Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Part. Diff. Equation., 18 (2002), pp. 584–608.CrossRefGoogle Scholar
[5]Kurganov, A. and Levy, D., Third-order semi-discrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22 (2000), pp. 1461–1488.Google Scholar
[6]Kurganov, A. and Petrova, G., Central schemes and contact discontinuities, Numer. Anal., 34 (2000), pp. 1259–1275.CrossRefGoogle Scholar
[7]Kurganov, A. and Petrova, G., A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., 88 (2001), pp. 683–729.Google Scholar
[8]Kurganov, A., Noelle, S. and Petrova, G., Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), pp. 707–740.CrossRefGoogle Scholar
[9]van Leer, B., Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes, Commun. Comput. Phys., 1(2) (2006), pp. 192–206.Google Scholar
[10]Shu, C.-W., High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD, Int. J. Comput. Fluid. Dyn., 17 (2003), pp. 107–118.Google Scholar
[11]Levy, D., Puppo, G. and Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22 (2000), pp. 656–672.Google Scholar
[12]Jiang, G. S., Levy, D., Lin, C. T., Osher, S. and Tadmor, E., High-resolution nonoscilla-tory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2147–2168.Google Scholar
[13]Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408–463.Google Scholar
[14]Shi, J., Zhang, Y.-T. and Shu, C.-W., Resolution of high order WENO schemes for complicated flow structures, J. Comput. Phys., 186 (2003), pp. 690–696.Google Scholar
[15] NVIDIA®, CUDA 2.0 Programming Guide. Available at .Google Scholar
[16] NVIDIA®, CUDA ZONE. Available at .Google Scholar
[17]Gottlieb, S. and Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math. Com-put., 67 (1998), pp. 73–85.Google Scholar
[18]Gottlieb, S., Shu, C. W. and Tadmor, E., Strong stability-preserving high order time discretization methods, SIAM Rev., 43 (2001), pp. 89–112.Google Scholar
[19]Brandvik, T. and Pullan, G., Acceleration of a 2D Euler flow solver using commodity graphics hardware, J. Mech. Eng. Sci., 221 (2007), pp. 1745–1748.Google Scholar
[20]Brandvik, T. and Pullan, G., Acceleration of a 3D Euler solver using commodity graphics hardware, 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008, pp. 607.Google Scholar
[21]Hagen, T. R., Lie, K.-A. and Natvig, J. R., Solving the Euler equations on graphics processing units, in Computational Science-ICCS 2006, Vol. 3994 of LNCS, Springer, 2006, pp. 220–227.Google Scholar
[22]Hagen, T. R., Henriksen, M. O., Hjelmervik, J. M. and Lie, K.-A., How to solve systems of conservation laws numerically using the graphics processor as a high-performance computational engine, in Geometric Modeling, Numerical Simulation and Optimization, Springer, 2007, pp. 211–264.Google Scholar