Magnetohydrodynamics of HEMP

Part of the Scientific Computation book series (SCIENTCOMP)


Using the continuum mechanics approach, details are presented for solving the equations of magnetohydrodynamics in two space dimensions and time. The problem considers cylindrical symmetry, in that only the H θ component of a magnetic field is present. The problem is formulated so that the stress contributions resulting from a magnetic field are incorporated into the stress tensor of an elastic-plastic computer program. In addition to the Lorentz force and magnetic diffusion, thermal and radiation diffusion are also treated. Presented here is the magnetohydrodynamic portion of the HEMP program including thermal and radiation diffusion. Details of the calculations are given for the case where only the H θ component of an applied magnetic field, H, is included. The problem consists of developing a finite difference approximation to the double operator, ∇ × ∇ × V, where V is a vector function. The mathematical problem is similar to that for approximating the double operator, ∇ ∙ ∇ V, where V is a scalar function. For the problem at hand, V is the magnetic field, H and V is the temperature, T. It is seen from the vector identity
$$\nabla \times \nabla \times H = \nabla (\nabla \cdot H) - \nabla \cdot \nabla H$$
$$\nabla \times \nabla \times H = - \nabla \cdot \nabla H = - {\nabla ^2}H,$$
$$\nabla \cdot H = 0.$$
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Difference Equation Zone Center Integration Path Radiation Diffusion Ohmic Heating 
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  1. [8.1]
    R. D. Richtmyer: Difference Methods for Initial Value Problems ( Interscience Publishers, New York, 1957 )zbMATHGoogle Scholar
  2. [8.2]
    J. M. LeBlanc: Lawrence Livermore Laboratory, private communication, June 1972Google Scholar
  3. [8.3]
    J. Chang: “An Introduction to Extrapolation Methods for Numerical Solution of Differential Equations”, Lawrence Livermore Laboratory, Report UCID-15992 (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  1. 1.Lawrence Livermore National LaboratoryLivermoreUSA

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