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1、Computational Magnetohydrodynamics Ravi Samtaney Princeton Plasma Physics Laboratory Princeton University samtaneypppl.gov ii c ? 2007 Ravi Samtaney ALL RIGHTS RESERVED Contents Prefacev 1Introduction to Magnetohydrodynamics1 1.1Magnetohydrodynamics- Where does it come from? . . . . . . .1 1.1.1MHD
2、Equations . . . . . . . . . . . . . . . . . . . . . .2 1.1.2 Single-fl uid Resistive MHD . . . . . . . . . . . . . . . .5 1.1.3Ideal MHD . . . . . . . . . . . . . . . . . . . . . . . . .6 1.1.4Extended MHD Models for Magnetized Plasmas . . . . .7 1.2Boundary Conditions. . . . . . . . . . . . . . . .
3、 . . . . . . .9 1.3Outline of this book . . . . . . . . . . . . . . . . . . . . . . . .9 2Finite Volume Methods11 2.1 Introduction to fi nite volume methods . . . . . . . . . . . . . .11 2.2Conservative vs. Non-conservative Formulation. . . . . . . . .12 2.2.1Godunov Symmetrization . . . . . . . . .
4、 . . . . . . . .13 2.3Seven vs. Eight Waves. . . . . . . . . . . . . . . . . . . . . .14 2.4Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . .16 2.4.1Linearized Riemann Solvers . . . . . . . . . . . . . . . .16 2.4.2Exact solution to the Riemann problem. . . . . . . . .17 2.4.3Degen
5、eracies . . . . . . . . . . . . . . . . . . . . . . . .18 2.5 Methods to ensure solenoidal magnetic fi elds. . . . . . . . . .18 2.6Constrained Transport Method . . . . . . . . . . . . . . . . . .21 2.7Unsplit Method for Ideal MHD . . . . . . . . . . . . . . . . . .24 2.7.1Algorithm Steps . . . . .
6、. . . . . . . . . . . . . . . . .25 2.8 Verifi cation tests . . . . . . . . . . . . . . . . . . . . . . . . . .27 2.8.1Linear wave propagation. . . . . . . . . . . . . . . . .28 2.8.2Circularly polarized Alfv en wave. . . . . . . . . . . . .28 2.8.3Shock-tube problem . . . . . . . . . . . . . . . .
7、. . . .29 2.8.4The Rotor problem. . . . . . . . . . . . . . . . . . . .29 iii ivCONTENTS 2.8.5MHD shock refraction . . . . . . . . . . . . . . . . . . .29 3Semi-implicit and Implicit Methods31 3.1Semi-implicit unsplit upwind method for compressible MHD. .31 3.1.1 Implicit treatment of diff usive fl
8、uxes . . . . . . . . . . .31 3.2Rationale for implicit treatment . . . . . . . . . . . . . . . . . .32 3.3Implicit treatment of fast compressional wave . . . . . . . . . .33 3.3.1A model illustration . . . . . . . . . . . . . . . . . . . .33 3.3.2Application to fast compressional wave . . . . . . .
9、. . .35 3.4Implicit treatment of the Alfv en wave. . . . . . . . . . . . . .37 3.5Fully implicit methods for MHD. . . . . . . . . . . . . . . . .38 3.5.1Implicit Upwind Method. . . . . . . . . . . . . . . . .38 3.5.2Introduction to Newton-Krylov method. . . . . . . . .41 3.5.2.1Preconditioners . . .
10、 . . . . . . . . . . . . . .43 3.5.3JFNK method for resistive MHD . . . . . . . . . . . . .44 3.5.3.1A model illustration . . . . . . . . . . . . . .44 3.5.3.2Application to resistive MHD . . . . . . . . . .45 3.5.4JFNK method for compressible MHD. . . . . . . . . .46 3.5.4.1Preconditioning Strategy
11、 . . . . . . . . . . . .48 3.6 Verifi cation Tests for Resistive MHD . . . . . . . . . . . . . . .50 3.6.1Magnetic Reconnection in 2D . . . . . . . . . . . . . . .50 4 Adaptive Mesh Refi nement51 4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 4.2Berger-Colella AMR for Hype
12、rbolic Systems . . . . . . . . . . .51 4.3Solving Elliptic Systems . . . . . . . . . . . . . . . . . . . . . .52 4.3.1MHD on AMR Grids. . . . . . . . . . . . . . . . . . .53 A Appendix I: Glossary of Acronyms55 B Bibliography57 Preface Computational fl uid dynamics (CFD) is a very interesting subjec
13、t which com- bines numerical analysis, algorithms, applied mathematics with the practical aim of solving fl uid fl ow problems by computational means. The motivations are many-fold: a desire to understand the physics of fl uid fl ow and turbu- lence, scientifi c and technological applications, as we
14、ll as a desire to peek in to the beautiful structure of solutions nonlinear partial diff erential equations. A large number of books have been written on CFD. On the other hand, there is precious little for computational plasma dynamics and magnetohydrodynam- ics. Magnetohydrodynamics is the “simple
15、st” model for plasma dynamics but it is, nonetheless, rich in mathematical structure and no less important in the modeling of several physical phenomenon ranging from MHD turbulent, stellar dynamics, and modeling at the device-scale of fusion devices such as tokamaks and stellarators.The focus of th
16、is book is computational magnetohydrody- namics which, in a sense, is a superset of computational gas dynamics because it combines the equations of hydrodynamics with Maxwells electromagnetic equations. There is a rather broad spectrum for MHD, too. On one end of the spectrum is ideal MHD which is a system of hyperbolic conservation laws, followed by resistive MHD, two-fl uid and Hall MHD, and extended MHD. The latter systems are dispersive, generally obeying a quadratic dispersion relation,
