An essentially nonoscillatory high-order Padé-type (ENO-Padé) scheme. (English) Zbl 1063.76070

A new high-order method for conservation laws is proposed, which combines the explicit ENO (essentially non-oscillatory) interpolation algorithm with the ideas of implicit cell-centered Padé schemes. The resulting linearly implicit method, called ENO-Padé scheme, is designed to eliminate the non-physical behavior of Padé schemes across discontinuities and to improve the accuracy of ENO schemes in smooth regions. Numerical experiments are presented for 2D scalar transport equations as well as for 1D and 2D compressible flows, and the new method are compared to other popular high-order schemes.


76M20 Finite difference methods applied to problems in fluid mechanics
76R99 Diffusion and convection
76N15 Gas dynamics (general theory)
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[1] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006
[2] A. G. Kravchenko, P. Moin, and, K. Shariff, B-Spline Method and Zonal Grids for Simulations of Complex Turbulent Flows, AIAA Paper 97-0433, 35th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1997.
[3] A. G. Kravchenko, and, P. Moin, B-Spline Methods and Zonal Grids for Numerical Simulations of Turbulent Flows, Report No. TF-73, Department of Mechanical Engineering, Stanford University, 1998.
[4] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, J. comput. phys., 31, 335, (1979) · Zbl 0416.76002
[5] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[6] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[7] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[8] C.-W. Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, NASA/CR-97-206253, 1997.
[9] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially nonoscillatory schemes, J. comput. phys., 115, 200, (1994)
[10] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[11] Rai, M.M.; Moin, P., Direct simulations of turbulent flow using finite-difference schemes, J. comput. phys., 96, 15, (1991) · Zbl 0726.76072
[12] P. G. Huang, Computation of turbulent flows using upwind-biased 5th-order schemes, Proceeding of the Eleventh Symposium on Turbulent Shear Flows, Vol. 2, 2-83, Grenoble, France, September 8-10, 1997.
[13] K. Mahesh, P. Moin, and, S. K. Lele, The Interaction of a Shock Wave with a Turbulent Shear Flow, Report No. TF-69, Department of Mechanical Engineering, Stanford University, 1997. · Zbl 0899.76193
[14] Deng, X.; Maekawa, H., Compact high-order accurate nonlinear schemes, J. comput. phys., 130, 77, (1997) · Zbl 0870.65075
[15] M. R. Visbal, and, D. V. Gaitonde, High-Order Accurate Methods for Unsteady Vortical Flows on Curvilinear Meshes, AIAA Paper 98-0131, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1998.
[16] D. V. Gaitonde, and, M. R. Visbal, Further Development of a Navier-Stokes Solution Procedure Based on Higher-Order Formulas, AIAA Paper 99-0557, 37th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1999.
[17] Cockburn, B.; Shu, C.-W., Nonlinearly stable compact schemes for shock calculations, SIAM J. numer. anal., 31, 607, (1994) · Zbl 0805.65085
[18] Yee, H.C., Explicit and implicit multidimensional compact high-resolution shock-capturing methods: formulation, J. comput. phys., 131, 216, (1997) · Zbl 0889.76054
[19] Adams, N.A.; Shariff, K., A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. comput. phys., 127, 27, (1996) · Zbl 0859.76041
[20] Fatemi, E.; Jerome, J.; Osher, S., Solution of the hydrodynamic device model using high order non-oscillatory shock capturing algorithms, IEEE trans. comput. aided design integrated circuits syst., 10, 232, (1991)
[21] Shu, C.-W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. sci. comput., 5, 127, (1990) · Zbl 0732.65085
[22] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[23] Rogerson, A.M.; Meiburg, E., A numerical study of the convergence properties of ENO schemes, J. sci. comput., 5, 151, (1990) · Zbl 0732.65086
[24] McKenzie, J.F.; Westphal, K.O., Phys. fluids, 11, 2350, (1968)
[25] H. C. Yee, Upwind and Symmetric Shock Capturing Schemes, NASA TM 89464, 1987.
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