An accurate asymptotic approximation and precise numerical solution of highly sensitive Troesch’s problem

摘要

Troesch’s problem is a two-point boundary value problem that arises in the confinement of a plasma column by radiation pressure and in the theory of gas porous electrodes. It has attracted significant amount of interest for the last 50 years. Current numerical solvers developed so far suffers significantly for large values of the sensitivity parameter λ. Here, we present a new scheme that can be shown to outperform every alternative numerical evaluation procedure and can provide accurate solutions for arbitrarily large values of the parameter. Our solution based on derivation of a new asymptotic form where the difference of solution and this approximation decays exponentially. For moderate values of the parameter, further, we use a discontinuous Galerkin method to calculate this difference. We include a variety of numerical results that confirm that, indeed, our algorithm compares favorably with alternative methods.