Summary
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).
RIACS Technical Report 00.11, June 2000, NASA Ames Research Center; Partial of this work was carried out while the second, third and fourth authors were visiting scientists with RIACS, NASA Ames Research Center
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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 ).
I. Daubechies (1992), Ten Lectures on Wavelets,CBMS-NSF Regional Conference Series in Applied Mathematics, No 61, SIAM.
M. Farge (1992), “Wavelet Transforms and their Applications to Turbulence,” Ann. Rev. of Fluid Mech., 24, 395–457.
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.
M. Gerritsen and P. Olsson (1996), “Designing an Efficient Solution Strategy for Fluid Flows: I. A Stable High Order Finite Difference Scheme and Sharp Shock Resolution for the Euler Equations,” J. Comput. Phys. 129, 245.
A. Harten (1978), “The Artificial Compression Method for Computation of Shocks and Contact Discontinuities: III. Self-Adjusting Hybrid Schemes,” Math. Comp. 32, 363.
A. Harten (1983), “On the Symmetric Form of Systems for Conservation Laws with Entropy,” ICASE Report 81–34, NASA Langley Research Center, 1981 also, J. Comput. Phys. 49, 151.
A. Harten (1995), “Multiresolution Algorithms for the Numerical Solution of Hyperbolic Conservation Laws,” Comm. Pure Appl. Math., 48, 1305–1342
S. Mallat and S. Zhong (1992), “Characterization of Signals from Multiscale Edges,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 710–732.
P. Olsson, (1995) “Summation by Parts, Projections and Stability,” Math. Comp. 64, 1035.
P. Olsson (1995), “Summation by Parts, Projections and Stability II.,” Math. Comp. 64, 212.
P. Olsson (1995), “Summation by Parts, Projections and Stability III,” RIACS Technical Report 95–06, NASA Ames Research Center.
P. Olsson and J. Oliger (1994), “Energy and Maximum Norm Estimates for Nonlinear Conservation Laws,” RIACS Technical Report 94–01, NASA Ames Research Center.
V. Perrier, T. Philipovitch and C. Basdevant (1999), “Wavelet Spectra Compared to Fourier Spectra,” Publication of ENS, Paris.
P.L. Roe (1981), “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys. 43, 357.
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.
C.W. Shu, “Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws,” NASA/CR-97–206253, 1997.
B. Sjögreen, “Numerical Experiments with the Multiresolution Scheme for the Compressible Euler Equations,” J. Comput. Phys., 117, 251–261 (1995).
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.
J.L. Steger and R.F. Warming, (1981) “Flux Vector Splitting of Inviscid Gas Dynamics Equations with Applications to Finite Difference Methods,” J. Comput. Phys. 40, 263.
B. Strand (1994), “Summation by Parts for Finite Difference Approximations for d/dx,” J. Comput. Phys. 110, 47.
M. Vinokur and H.C. Yee, (2000) “Extension of Efficient Low Dissipative High Order
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, CA
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.
H.C. Yee, G.H. Klopfer and J.-L. Montagne (1990), “High-Resolution Shock-Capturing Schemes for Inviscid and Viscous Hypersonic Flows,” J. Comput. Phys., 88, 31.
H.C. Yee, N.D. Sandham and M.J. Djomehri (1999a), “Low Dissipative High Order Shock-Capturing Methods Using Characteristic-Based Filters,” RIACS Technical Report 98.11, May 1998, also, J. Comput. Physics, 150, 199–238.
H.C. Yee, M. Vinokur and M.J. Djomehri (1999b). Djomehri (1999b), “Entropy Splitting and Numerical Dissipation,” NASA Technical Memorandum 208793, August, 1999, NASA Ames Research Center, also, J. Comput. Physics, 162, 33–81, 2000.
H.C. Yee, M. Vinokur and M.J. Djomehri (2000). Djomehri (2000), “Entropy Splitting and Numerical Dissipation,” J. Comput. Phys., 162, 33–81, 2000.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yee, H.C., Sjögreen, B., Sandham, N.D., Hadjadj, A. (2001). Progress in the Development of a Class of Efficient Low Dissipative High Order Shock-Capturing Methods. In: Computational Fluid Dynamics for the 21st Century. Notes on Numerical Fluid Mechanics (NNFM), vol 78. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44959-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-44959-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07558-2
Online ISBN: 978-3-540-44959-1
eBook Packages: Springer Book Archive