a new time treatment for solving partial integro-differential

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

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IMA Journal ofNumericalAnalysis

Volume 19Issue 1

Pp. 91-103.

A new t ime t r ea tm en t f o r so lving pa r t i a l i n t eg r o - d if fe r en t i a le q u a t i o n s o f r a d i a t iv e t r a n s p o r t

JI Fran kel and GE Osborne

1.Mechanical and Aerospace Engineering and Engineering Sciences Department, University ofTennessee, Knoxville, TN 37966- 2210 , USA

Abstract

A new nonmarching time treatment is offered for solving nonlinear weakly singular, partial integro-differential equations of the type that appear in transient, conductive and radiative transport. Unlikeconventional methods that involve time marching for solving transient problems, the proposed conceptsimultaneously resolves the entire space- time domain. The context invol ves transient cooling of a mediumin which volum etric radiative effects are present. This example ill ustrates the methodology and i ts salientfeatures without und ue complication to either the physics or mathematics. The approach begins byreformulating the original mathematical description into a form conducive to collocation. Cumulativevariables are developed in order t o readily apply th e recently proposed method of Kumar & Sloan. Thisformalism allows for the efficient utilization of collocation in both space and time. The expansions for theunknown f unctions of i nterest are expressed in term s of global basis functi ons composed of Chebyshevpolynomials of the first kind. The proposed method illustrates that a nonmarching temporal treatmentproduces accurate and stable numerical results.

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1.IMA J Numer Anal (1999) 19 (1): 91-103.

doi: 10.1093/imanum/19.1.91

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