Non-linear filtering and limiting in high order methods for ideal and non-ideal MHD

The adaptive nonlinear filtering and limiting in spatially high order schemes (Yee et al. J. Comput. Phys. 150, 199 - 238, ( 1999), Sjogreen 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, (Sjogreen and Yee, ( 2003), Proceedings of the 16th AIAA/CFD conference, June 23 - 26, Orlando F1; Yee and Sjogreen ( 2003), Proceedings of the International Conference on High Performance Scientific Computing, March, 10 - 14, Honai, Vietnam; Yee and Sjogreen ( 2003), RIACS Technical Report TR03. 10, July, NASA Ames Research Center; Yee and Sjogreen ( 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.