High accuracy compact schemes and Gibbs’ phenomenon. (English) Zbl 1071.76040
Summary: Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme [T. K. Sengupta et al., J. Comp. Phys. 192, No. 2, 677–694 (2003; Zbl 1038.65082)] is compared with another from the literature [X. Zhong, J. Comp. Phys. 144, 622–709 (1998)] that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third-order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs’ phenomenon of the solution. The essential causes of Gibbs’ phenomenon also explained.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76D99 | Incompressible viscous fluids |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Citations:
Zbl 1038.65082References:
| [1] | Lele S. K. (1992). Compact finite difference schemes with spectral like resolution. J. Comp. Phys. 103, 16-42. · Zbl 0759.65006 · doi:10.1016/0021-9991(92)90324-R |
| [2] | Haras Z., and Ta ’asan S. (1994). Finite difference schemes for long-time integration. J. Comput. Phys. 114, 265-279. · Zbl 0808.65083 · doi:10.1006/jcph.1994.1165 |
| [3] | Adams N. A., and Shariff K. (1996). A high resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. J. Comp. Phys. 127, 27-51. · Zbl 0859.76041 · doi:10.1006/jcph.1996.0156 |
| [4] | Zhong X. (1998). High order nite difference schemes for numerical simulation of hypersonic boundary layer transition. J. Comp. Phys. 144, 622-709. · Zbl 0935.76066 · doi:10.1006/jcph.1998.6010 |
| [5] | Sengupta T. K., Ganerwal G., and De S. (2003). Analysis of central and compact schemes. J. Comp. Phys. 192(2), 677-694. · Zbl 1038.65082 · doi:10.1016/j.jcp.2003.07.015 |
| [6] | Tam C. K. W., and Webb J. C. (1993). Dispersion-relation-preserving nite difference schemes for computational acoustics. J. Comp. Phys. 107, 262-281. · Zbl 0790.76057 · doi:10.1006/jcph.1993.1142 |
| [7] | Lighthill M. J. (1978). Waves in Fluids, Cambridge University Press, U. K. · Zbl 0375.76001 |
| [8] | Sengupta T. K., Sridar D., Sarkar S., and De S. (2001). Spectral analysis of ux vector splitting nite volume methods. Int. J. Numer. Meth. Fluids. 37, 149-174. · Zbl 0996.76064 · doi:10.1002/fld.168 |
| [9] | Jiang G. S and Shu C.-W. Efficient implementation of weighted ENO schemes. J. Comp. Phys. 126, 202. · Zbl 0877.65065 |
| [10] | Pirozzoli S. (2002). Conservative hybrid compact-WENO scheme for shock-turbulence interaction. J. Comp. Phys. 178, 81-117. · Zbl 1045.76029 · doi:10.1006/jcph.2002.7021 |
| [11] | van Leer B. (1974). Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comp. Phys. 14, 361-370. · Zbl 0276.65055 · doi:10.1016/0021-9991(74)90019-9 |
| [12] | Roe P. L., and Baines M. J. (1982). Algorithms for advection and shock problems. In the Proc. of 4th GAMM Conf. on Numerical Methods in Fluid Mechanics. Vieweg, Braunschweig. · Zbl 0482.76008 |
| [13] | Roe P. L. (1985). Some contribution to the modelling of discontinuous flows. Lectures in Applied Mathematics, 22, 163-193. |
| [14] | van Leer B. (1979). Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov ’s method. J. Comp. Phys. 32, 101-136. · Zbl 1364.65223 · doi:10.1016/0021-9991(79)90145-1 |
| [15] | Koren B. (1993). A robust upwind discretization method for advection, diffusion and source term. In Numerical Methods for Advection-Diffusion Problems (C. B. Vreugdenhil and B. Koren, eds. ), Vieweg, braunschweig. pp. 117-137. · Zbl 0805.76051 |
| [16] | Sengupta T. K., Anuradha G., and De S. (2003). Incompressible Navier-Stokes solution by new compact schemes. In Proc. of 2nd MIT Conf. on Comp. Fluid and Solid Mech. (K. J. Bathe, ed., ). |
| [17] | Lax P. D., and Wendroff B. (1960). Systems of conservation laws. Comm. Pure Appl. Math. 13, 217-237. · Zbl 0152.44802 · doi:10.1002/cpa.3160130205 |
| [18] | Harten A., Engquist B., Osher S., and Chakravarthy S. (1987). Uniformly high order essentially non oscillatory schemes III. J. Comp. Phys. 71, 213. · Zbl 0652.65067 · doi:10.1016/0021-9991(87)90031-3 |
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