Finite Difference Scheme
To arrive at finite difference equations modeling magnetohydrodynamic equilibrium we use a technique that is motivated by the finite element method . First we develop a second order accurate numerical quadrature formula for the Hamiltonian E based on a rectangular grid of mesh points over a unit cube of the space with coordinates s, u and v. We differentiate that formula with respect to nodal values of the unknowns R, ψ r0 and z0 in order to derive difference approximations to the magnetostatic equations. This procedure yields equations in a conservation form that is automatically compatible with conditions stemming from the fact that the flux function ψ is only determined up to an arbitrary additive function of s. Thus we avoid elementary difficulties with existence of solutions of the equilibrium problem which tend to obscure the more serious issues raised by the KAM theory.
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