A new time treatment for solving partial integro-differential equations ofradiative transport

A new nonmarching time treatment is offered for solving nonlinear weakly si ngular, partial integro-differential equations of the type that appear in t ransient, conductive and radiative transport. Unlike conventional methods t hat involve time marching for solving transient problems, the proposed conc ept simultaneously resolves the entire space-time domain. The present conte xt involves transient cooling of a medium in which volumetric radiative eff ects are present. This example illustrates the methodology and its salient features without undue complication to either the physics or mathematics. T he approach begins by reformulating the original mathematical description i nto a form conducive to collocation. Cumulative variables are developed in order to readily apply the recently proposed method of Kumar & Sloan. This formalism allows for the efficient utilization of collocation in both space and time. The expansions for the unknown functions of interest are express ed in terms of global basis functions composed of Chebyshev polynomials of the first kind. The proposed method illustrates that a nonmarching temporal treatment produces accurate and stable numerical results.