R. E. Grundy

Exact solutions for reconnective magnetic annihilation

A family of exact solutions of the steady resistive nonlinear magnetohydrodynamic equations in two dimensions (x, y) is presented for reconnective annihilation, in which the magnetic field is advected across one pair of separatrices and diffuses across the other pair. They represent a two-fold generalization of the previous Craig-Henton solution, since a dimensionless free parameter (γ) in the new solutions equals unity in the previous solutions and the components (vxe, vye) and (Bxe, Bye) of plasma velocity and magnetic field at a fixed external point (x, y) = (1, 0), say, may all be imposed, whereas only three of these four components are free in the previous solutions. The solutions have the exact forms A= A 0 (x)+ A 1 (x)y,ψ= ψ 0 (x)+ ψ 1 (x)y for the magnetic flux function (A) and stream function (ψ), so that the electric current is no longer purely a function of x as it was previously. The origin (0,0) represents both a stagnation point and a magnetic null point, where the plasma velocity (v = ∇ × ψ) and magnetic field (B = ∇ × ψ) both vanish. A current sheet extends along the y-axis. The nonlinear fourth-order equations for A1 and ψ1 are solved in the limit of small dimensionless resistivity (large magnetic Reynolds number) using the method of matched asymptotic expansions. Although the solution has a weak boundary layer near x = 0, we show that a composite asymptotic representation on 0 ⩽ x⩽ 1 is given by the leading-order outer solution, which has a simple closed-form structure. This enables the equations for A0 and ψ0 to be solved explicitly, from which their representation for small resistivity is obtained. The effect of the five parameters (vxe, vye, Bxe, Bye, γ) on the solutions is determined, including their influence on the width of the diffusion region and the inclinations of the streamlines and magnetic field lines at the origin. Several possibilities for generalizing these solutions for asymmetric reconnective annihilation in two and three dimensions are also presented.

The analysis of initial-boundary-value problems using Hermite interpolation

In this paper we conduct an investigation into the feasibility of using Hermite interpolation as a practical tool for constructing polynomial approximations to initial-boundary-value problems for partial differential equations. The semi-analytic method uses Hermite interpolants to systematically estimate the time-dependent end point function values and/or derivatives which are not given by the boundary conditions or determined by the equations themselves. These estimates can then be used to analyse both the qualitative and quantitative structure of solutions. The idea is introduced via a series of examples intended to highlight various aspects of the method. These include a number of diffusion-convection-reaction models and examples involving unknown moving boundaries which illustrate how we can use the technique to identify such features as blow up regimes, steady states and similarity solutions together with their stability properties.

Hermite interpolation visits ordinary two-point boundary value problems

This paper is concerned with constructing polynomial solutions to ordinary boundary value problems. A semi-analytic technique using two-point Hermite interpolation is compared with conventional methods via a series of examples and is shown to be generally superior, particularly for problems involving nonlinear equations and/or boundary conditions.