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

A new nonmarching time treatment is offered for solving nonlinearweakly singular, partial integro-differential equations of thetype that appear in transient, conductive and radiative transport.Unlike conventional methods that involve time marching for solvingtransient problems, the proposed concept simultaneously resolvesthe entire space-time domain. The context involves transientcooling of a medium in which volumetric radiative effects arepresent. This example illustrates the methodology and its salientfeatures without undue complication to either the physics ormathematics. The approach begins by reformulating the originalmathematical description into a form conducive to collocation.Cumulative variables are developed in order to readily applythe recently proposed method of Kumar & Sloan. This formalismallows for the efficient utilization of collocation in bothspace and time. The expansions for the unknown functions ofinterest are expressed in terms of global basis functions composedof Chebyshev polynomials of the first kind. The proposed methodillustrates that a nonmarching temporal treatment produces accurateand stable numerical results.