Abstract
In many plasma physical and astrophysical problems, both linear and nonlinear effects can lead to global dynamics that induce, or occur simultaneously with, local phenomena. For example, a magnetically confined plasma column can potentially posses global magnetohydrodynamic (MHD) eigenmodes with an oscillation frequency that matches a local eigenfrequency at some specific internal radius. The corresponding linear eigenfunctions then demonstrate large-scale perturbations together with fine-scale resonant behaviour. A well-known nonlinear effect is the steepening of waves into shocks where the discontinuities that then develop can be viewed as extreme cases of ‘short wavelength’ features. Numerical simulations of these types of physics problems can benefit greatly from dynamically controlled grid adaptation schemes.
Here, we present a progress report on two different approaches that we envisage to evaluate against each other and use in multi-dimensional hydro- and magnetohydrodynamic computations. In r-refinement, the number of grid points stays fixed, but the grid ‘moves’ in response to persistent or developing steep gradients. First results on 1D and 2D MHD model problems are presented. In h-refinement, the resolution is raised locally without moving individual mesh points. We show 2D hydrodynamic ‘shock tube’ evolutions where hierarchically nested patches of subsequently finer grid spacing are created and destroyed when needed. This adaptive mesh refinement technique will be further implemented in the Versatile Advection Code, so that its functionality carries over to any set of near conservation laws in one, two, or three space dimensions.
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References
Berger, M.J.: Data structures for adaptive grid generation, SIAM J. Sci. Stat. Comput. 7(3), 904 (1986)
Boris, J.P., Book, D.L.: Flux-corrected transport. I.SHASTA, A fluid transport algorithm that works, J. Comput. Phys. 11, 38 (1973)
De Sterck, H., Low, B.C., Poedts, S.: Characteristic analysis of a complex two-dimensional magnetohydrodynamic bow shock flow with steady compound shocks, Phys. of Plasmas 6, 954 (1999)
Dorfi, E.A., Drury, L. O’C.: Simple adaptive grids for 1-D initial value problems, J. Comput. Phys. 69, 175 (1987)
Friedel, H., Grauer, R., Marliane, C.: Adaptive mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows, J. Comput. Phys. 134, 190–198 (1997)
Harten, A.: High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49, 357 (1983)
Keppens, R., Tóth, G.: Simulating Magnetized Plasmas with the Versatile Advection Code, in VECPAR’98-Third International Conference for Vector and Parallel Processing, Lecture Notes in Computer Science, 1573, edited by J. M. L. M. Palma, J. Dongarra and V. Hernandez p. 680–690 (Springer-Verlag, 1999)
Keppens, R., Tóth, G., Westermann, R.H.J., Goedbloed, J.P.: Growth and saturation of the Kelvin-Helmholtz instability with parallel and anti-parallel magnetic fields, J. Plasma Phys. 61, 1 (1999)
Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T. I., De Zeeuw, D.L.: A Solution-Adaptive Upwind Scheme for Ideal Magnetohydrodynamics, J. Comput. Phys. 154, 284–309 (1999)
Steiner, O., Knölker, M., Schüssler, M.: Dynamic interaction of convection with magnetic flux sheets: first results of a new MHD code, in Proc. NATO advanced research workshop ASI Series C-433, Solar Surface Magnetism, edited by R.J. Rutten and C.J. Schrijver, p. 441–470 (Kluwer Dordrecht, 1994)
Stenuit, H., Keppens, R., Goossens, M.: Eigenfrequencies and optimal driving frequencies of 1D non-uniform magnetic flux tubes, Astron. & Astrophys. 331, 392 (1998)
Tóth, G.: Versatile Advection Code, in Proceedings of High Performance Computing and Networking Europe 1997, Lecture Notes in Computer Science, 1225, edited by B. Hertzberger and P. Sloot, p. 253–262 (Springer-Verlag, 1997)
Tóth, G.: The LASY Preprocessor and its Application to General Multi-Dimensional Codes, J. Comput. Phys. 138, 981 (1997)
Tóth, G., Keppens, R.: Comparison of Different Computer Platforms for Running the Versatile Advection Code, in Proceedings of High Performance Computing and Networking Europe 1998, Lecture Notes in Computer Science, 1401, edited by P. Sloot, M. Bubak and B. Hertzberger p. 368–376 (Springer-Verlag, 1998)
Tóth, G., Keppens, R., Botchev, M. A.: Implicit and semi-implicit schemes in the Versatile Advection Code: numerical tests, Astron. & Astrophys. 332, 1159 (1998)
Weiss, N.O.: The expulsion of magnetic flux by eddies, Proc. Roy. Soc. A 293, 310 (1966)
Zegeling, P.A.: r-refinement for evolutionary PDEs with finite elements or finite differences, Applied Numer. Math. 26, 97 (1998)
Zegeling, P.A., Keppens, R.: Adaptive Method of Lines for Magneto-Hydrodynamic PDE Models, in preparation.
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Keppens, R., Nool, M., Zegeling, P.A., Goedbloed, J.P. (2000). Dynamic Grid Adaptation for Computational Magnetohydrodynamics. In: Bubak, M., Afsarmanesh, H., Hertzberger, B., Williams, R. (eds) High Performance Computing and Networking. HPCN-Europe 2000. Lecture Notes in Computer Science, vol 1823. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45492-6_7
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DOI: https://doi.org/10.1007/3-540-45492-6_7
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