High order hybrid central-WENO finite difference scheme for conservation laws. (English) Zbl 1115.65091
Summary: We present a high resolution hybrid central finite difference-WENO scheme for the solution of conservation laws, in particular, those related to shock-turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory (WENO) scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
Keywords:
central finite difference; multi-resolution; conservation laws; finite difference-WENO scheme; shock-turbulence interaction; weighted essentially non-oscillatory scheme; numerical experiments; compressible Euler equationsReferences:
| [1] | Adams, N.; Shariff, K., High-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys., 127, 27-51 (1996) · Zbl 0859.76041 |
| [2] | Balsara, D.; Shu, C. W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452 (2000) · Zbl 0961.65078 |
| [3] | Bihari, B.; Harten, A., Multiresolution schemes for the numerical solution of 2D Conservation Laws, SIAM J. Sci. Comput., 18, 2, 315-354 (1997) · Zbl 0878.35007 |
| [4] | Bogey, C.; Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, J. Comput. Phys., 194, 194-214 (2004) · Zbl 1042.76044 |
| [5] | Choen, A.; Kaber, S. M.; Muller, S.; Postel, M., Fully adaptive multiresolution finite volume schemes for conservation laws, Math. Comput., 72, 241, 183-225 (2001) · Zbl 1010.65035 |
| [6] | Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conferences Series in Applied Mathematics (1992), SIAM, Capital City Press: SIAM, Capital City Press Montpellier, Vermont · Zbl 0776.42018 |
| [7] | Harten, A., High resolution schemes for hyperbolic conservation laws, Comput. Phys., 49, 357-393 (1983) · Zbl 0565.65050 |
| [8] | Harten, A., Adaptive multiresolution schemes for shock computations, Comput. Phys., 115, 319-338 (1994) · Zbl 0925.65151 |
| [9] | Henrick, A. K.; Aslam, T. D.; Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207, 542-567 (2005) · Zbl 1072.65114 |
| [10] | Hill, D.; Pullin, D., Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks, J. Comput. Phys., 194, 435-450 (2004) · Zbl 1100.76030 |
| [11] | Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228 (1996) · Zbl 0877.65065 |
| [12] | Pirozzoli, S., Conservative hybrid compact-WENO schemes for shock-turbulence interaction, J. Comput. Phys., 178, 1, 81-117 (2002) · Zbl 1045.76029 |
| [13] | C.W. Shu, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325-432.; C.W. Shu, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325-432. · Zbl 0904.00047 |
| [14] | Shu, C. W.; Erlebacher, G.; Zang, T.; Whitaker, D.; Osher, S., High order ENO schemes applied to two- and three-dimensional compressible flow, Appl. Numer. Math., 9, 45-71 (1992) · Zbl 0741.76052 |
| [15] | Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83, 1, 32-78 (1989) · Zbl 0674.65061 |
| [16] | Vasilyev, O. V.; Lund, T. S.; Moin, P., A general class of commutative filters for LES in complex geometries, J. Comput. Phys., 146, 82-104 (1998) · Zbl 0913.76072 |
| [17] | Zingg, D. W., Comparison of high-accuracy finite difference methods for linear wave propagation, SIAM J. Sci. Comput., 22, 2, 476-502 (2000) · Zbl 0968.65061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
