Computational Methods in Plasma Physics by Stephen Jardin

By Stephen Jardin

Assuming no previous wisdom of plasma physics or numerical tools, Computational equipment in Plasma Physics covers the computational arithmetic and strategies had to simulate magnetically restrained plasmas in sleek magnetic fusion experiments and destiny magnetic fusion reactors. mostly self-contained, the textual content offers the fundamental recommendations worthwhile for the numerical resolution of partial differential equations. besides discussing numerical balance and accuracy, the writer explores a number of the algorithms used this day in adequate intensity in order that readers can learn their balance, potency, and scaling homes. He specializes in mathematical types the place the plasma is taken care of as a carrying out fluid, in view that this can be the main mature plasma version and such a lot acceptable to experiments. The booklet additionally emphasizes toroidal confinement geometries, rather the tokamak—a very profitable configuration for confining a high-temperature plasma. the various uncomplicated numerical thoughts provided also are acceptable for equations encountered in a higher-dimensional section house. probably the most difficult learn parts in glossy technological know-how is to boost compatible algorithms that bring about sturdy and actual recommendations which can span proper time and house scales. This publication offers a good operating wisdom of the algorithms utilized by the plasma physics neighborhood, supporting readers on their approach to extra complicated learn.

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On a characteristic manifold, relations exist between v , B , p , s . 22 Computational Methods in Plasma Physics Discontinuities satisfying these relations are propagated along with the characteristics. Substitution of the roots of Eq. 92) into the matrix in Eq. 91) yields the following eigenvectors: Entropy             ρvx ρvy ρvz ρ/µ0 Bx ρ/µ0 By ρ/µ0 Bz 1 cS p s Magnetoacoustic Alfv´en       0 0 nx u2 /(u2 − VA2 )    0   1   0         0   0   n z    =      0  ,  0  ,  −nx uVA nz /(u2 − V 2 ) A         0   ±1   0         0   0   n2x uVA /(u2 − V 2 ) A        0   0   u/cS 0 ±1 0       .

68) over a volume and using Gauss’s theorem to convert divergences to surface integrals over the boundary. 71) 12 Computational Methods in Plasma Physics ∂ ∂t − 1 2 3 B2 ρu + p + 2 2 2µ0 1 2 ρu + dSˆ n· u 2 d3 x = 5 p 2 + u·π+ 1 E × B + q . 72) µo We note in passing that Stokes’ theorem can also be applied directly to any surface integral of Eq. 24) to obtain the flux conservation relation in a more familiar form. Also, if the resistivity is zero so that E + u × B = 0, then this implies that the electric field in the frame of the fluid is zero, and a conservation law can be used to show that the magnetic flux in any fluid element is conserved.

4) ∂t ∂x ∂x In the finite difference approximation, the exact solution, φ, is approximated by a discrete state vector, φ∆ , which is defined only on integer spacetime points, thus φ ⇒ φ∆ , L= and the exact operator L gets replaced by a finite difference operator L ⇒ L∆ , giving the finite difference approximation to the partial differential equation L∆ {φ∆ } = 0. There are two sources of error in obtaining numerical solutions, and these behave quite differently. These are (i) truncation error T∆ ≡ L∆ {φ} − L{φ}, Introduction to Finite Difference Equations 31 and (ii) machine error, or round-off error.

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