Progress in the Development of a Class of Efficient Low Dissipative High Order Shock-Capturing Methods
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In a series of papers, Olsson (1994, 1995), Olsson & Oliger (1994), Strand (1994), Gerritsen & Olsson (1996), Yee et al. (1999a,b, 2000) and Sandham & Yee (2000), the issue of nonlinear stability of the compressible Euler and Navier-Stokes Equations, including physical boundaries, and the corresponding development of the discrete analogue of nonlinear stable high order schemes, including boundary schemes, were developed, extended and evaluated for various fluid flows. High order here refers to spatial schemes that are essentially fourth-order or higher away from shock and shear regions. The objective of this paper is to give an overview of the progress of the low dissipative high order shock-capturing schemes proposed by Yee et al. (1999a,b, 2000). This class of schemes consists of simple non-dissipative high order compact or non-compact central spatial differencings and adaptive nonlinear numerical dissipation operators to minimize the use of numerical dissipation. The amount of numerical dissipation is further minimized by applying the scheme to the entropy splitting form of the inviscid flux derivatives, and by rewriting the viscous terms to minimize odd-even decoupling before the application of the central scheme (Sandham & Yee).
The efficiency and accuracy of these schemes are compared with spectral, TVD and fifth-order WENO schemes. A new approach of Sjögreen & Yee (2000) utilizing non-orthogonal multi-resolution wavelet basis functions as sensors to dynamically determine the appropriate amount of numerical dissipation to be added to the non-dissipative high order spatial scheme at each grid point will be discussed. Numerical experiments of long time integration of smooth flows, shock-turbulence interactions, direct numerical simulations of a 3-D compressible turbulent plane channel flow, and various mixing layer problems indicate that these schemes are especially suitable for practical complex problems in nonlinear aeroacoustics, rotorcraft dynamics, direct numerical simulation or large eddy simulation of compressible turbulent flows at various speeds including high-speed shock-turbulence interactions, and general long time wave propagation problems. These schemes, including entropy splitting, have also been extended to freestream preserving schemes on curvilinear moving grids for a thermally perfect gas (Vinokur & Yee 2000).
KeywordsNumerical Dissipation Density Contour WENO Scheme Total Variation Diminish Wavelet Basis Function
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- M.H. Carpenter, J. Nordstrom and D. Gottlieb (1998), “A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy,” NASA/CR-1998–206921 (also ICASE Report 98–12 ).Google Scholar
- I. Daubechies (1992), Ten Lectures on Wavelets,CBMS-NSF Regional Conference Series in Applied Mathematics, No 61, SIAM.Google Scholar
- A. Hadjadj, H.C. Yee and N.D. Sandham (2001), “Comparison of WENO with Low Dissipative High Order Schemes for Compressible Turbulence Computations”, RIACS Technical Report, NASA Ames Research Center.Google Scholar
- P. Olsson (1995), “Summation by Parts, Projections and Stability II.,” Math. Comp. 64, 212.Google Scholar
- P. Olsson (1995), “Summation by Parts, Projections and Stability III,” RIACS Technical Report 95–06, NASA Ames Research Center.Google Scholar
- P. Olsson and J. Oliger (1994), “Energy and Maximum Norm Estimates for Nonlinear Conservation Laws,” RIACS Technical Report 94–01, NASA Ames Research Center.Google Scholar
- V. Perrier, T. Philipovitch and C. Basdevant (1999), “Wavelet Spectra Compared to Fourier Spectra,” Publication of ENS, Paris.Google Scholar
- N.D. Sandham, and H.C. Yee (2000), “Entropy Splitting for High Order Numerical Simulation of Compressible Turbulence,” RIACS Technical Report 00.10, June 2000, NASA Ames Research Center; Proceedings of the First International Conference on CFD, July 10–14, 2000, Kyoto, Japan.Google Scholar
- C.W. Shu, “Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws,” NASA/CR-97–206253, 1997.Google Scholar
- B. Sjögreen and H.C. Yee (2000), “Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computations,” RIACS Technical Report 01.01, October, 2000, NASA Ames Research Center.Google Scholar
- B. Strand (1994), “Summation by Parts for Finite Difference Approximations for d/dx,” J. Comput. Phys. 110, 47.Google Scholar
- M. Vinokur and H.C. Yee, (2000) “Extension of Efficient Low Dissipative High OrderGoogle Scholar
- Schemes for 3-D Curvilinear Moving Girds,“ NASA Technical Memorandum 209598, June 2000, NASA Ames Research Center; Proceedings of Computing the Future III: Frontiers of CFD — 2000, June 26–28, 2000, Half Moon Bay, CAGoogle Scholar
- H.C. Yee (1989), “A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods,” VKI Lecture Series 1989–04 March 6–10, 1989, also NASA TM-101088, Feb. 1989.Google Scholar