Abstract
The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199–238, (1999), Sjögreen and Yee, J. Scient. Comput. 20, 211–255, (2004)) for the compressible Euler and Navier–Stokes equations have been recently extended to the ideal and non-ideal magnetohydrodynamics (MHD) equations, (Sjögreen and Yee, (2003), Proceedings of the 16th AIAA/CFD conference, June 23–26, Orlando F1; Yee and Sjögreen (2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10–14, Honai, Vietnam; Yee and Sjögreen (2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjögreen (2004), Proceedings of the ICCF03, July 12–16, Toronto, Canada). The numerical dissipation control in these adaptive filter schemes consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is very general. The objective of this paper is to investigate the performance of three commonly used types of discontinuity capturing nonlinear numerical dissipation for both the ideal and non-ideal MHD.
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References
Brio M., Wu C.C. (1988). An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400–422
Dai W., Woodward P.R. (1998). A simple finite difference scheme for multidimensional magnetohydrodynamical equations. J. Comput. Phys. 142, 331–369
Evans C.R., Hawley J.F. (1988). Simulation of magnetohydrodynamic flows: a constrained transport method. Astrophys. J. 332, 659–677
Gallice G. (1997). Systéme D’Euler-Poisson, Magnétohydrodynamique et Schemeas de Roe, PhD Thesis, L’Université Bordeaux I
Gaitonde, D. V. (1999). Development of a Solver for 3-D Non-Ideal Magnetogasdynamics AIAA Paper 99–3610
Godunov S.K. (1972). Symmetric form of the equations of magnetohydrodynamics, numerical methods for mechanics of continuum medium. 13(1): 26–34
Harten A., Hyman J.M. (1983). A self-adjusting grid for the computation of weak solutions of hyperbolic conservation laws. J. Comput. Phys. 50, 235–269
Jiang G.-S., Shu C.-W. (1996). Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228
Nessyahu H., Tadmor E. (1990). Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463
Nordstrom J., Carpenter M.H. (1999). Boundary and interface conditions for high-order finite-difference schemes applied to the Euler and Navier–Stokes equations. J. Comput. Phys. 148, 621–645
Olsson P. (1995). Summation by parts, projections and stability. I. Math. Comp. 64, 1035–1065
Powell, K. G. (1994). An Approximate Riemann Solver for Magnetohydrodynamics (That works in More than One Dimension), ICASE-Report 94-24, NASA Langley Research Center, April
Sjögreen B. (1995). High order centered difference methods for the compressible Navier–Stokes equations. J. Comput. Phys. 117, 67–78
Sjögreen, B., and Yee, H. C. (2004). Multiresolution wavelet based adaptive numerical dissipation control for shock-turbulence computation, RIACS Technical Report TR01.01, NASA Ames research center (Oct 2000); also, J. Scient. Comput. 20, 211–255
Sjögreen, B., and Yee, H. C. (2003). Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows, I: Basic Theory, AIAA 2003-4118, Proceedings of the 16th AIAA/CFD Conference, June 23–26, Orlando, Fl
Tóth, G. (2000). The div B=0 constraint in shock-capturing magnetohydrodynamic codes. J. Comput. Phys. 161, 605–652
Yee, H. C. (1989). A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods, VKI Lecture Series 1989–04, March 6–10, also NASA TM-101088, Feb. 1989
Yee H.C., Sandham N.D., Djomehri M.J. (1999). Low dissipative high order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150, 199–238
Yee, H. C., and Sjögreen, B. (2002). Designing Adaptive Low Dissipative High Order Schemes for Long-Time Integrations, Turbulent Flow Computation, (Eds. D. Drikakis & B. Geurts), Kluwer Academic Publisher; also RIACS Technical Report TR01-28, Dec. 2002
Yee, H. C., and Sjögreen, B. (2003). Divergence Free High Order Filter Methods for the Compressible MHD Equations, Proceedings of the International Conference on High Performance Scientific Computing, March 10–14, 2003, Hanoi, Vietnam
Yee, H. C., and Sjögreen, B. (2003). Efficient Low Dissipative High Order Scheme for Multiscale MHD Flows, II: Minimization of Div(B) Numerical Error, RIACS Technical Report TR03.10, July NASA Ames Research Center
Yee, H. C., and Sjögreen, B. (2004). Adaptive Numerical Dissipation Control in High Order Schemes for 3-D Non-Ideal MHD, Proceedings of the ICCFD3, July 12–16, Toronto, Canada
Yee, H. C., and Sjögreen, B. (2005). Nonlinear Filtering in Compact High Order Schemes, Proceedings of the ICNSP, July 12–15, Nara, Japan
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Yee, H.C., Sjögreen, B. Non-Linear Filtering and Limiting in High Order Methods for Ideal and Non-Ideal MHD. J Sci Comput 27, 507–521 (2006). https://doi.org/10.1007/s10915-005-9024-1
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DOI: https://doi.org/10.1007/s10915-005-9024-1