A MULTILEVEL METHOD FOR CONDUCTIVE-RADIATIVE HEAT TRANSFER.pdf

A MULTILEVEL METHOD FOR CONDUCTIVE-RADIATIVE HEATTRANSFER J. M. BANOCZIyAND C. T. KELLEYyWe present a fast multilevel algorithm for the solution of a system of nonlinearintegro-dierential equations that model steady-state combined radiative-conductiveheat transfer. The equations can be formulated as a compact xed point problem witha xed point map that requires both a solution of the linear transport equation andthe linear heat equation for its evaluation. We use fast transport solvers developedby the second author [12], [10], to construct an ecient evaluation of the xed pointmap and then apply the Atkinson-Brakhage [2], [3], method, with Newton-GMRESas the coarse mesh solver, to the full nonlinear system.1. Introduction. We consider the normalized dimensionless form of the equa-tions [17], [26], [27]. The radiative transport equation is@ @x (x; ) + (x; ) = c(x)2 Z 11 (x; 0) d0 + (1 c(x))4(x);(1)for x 2 (0;  ) with boundary conditions (0; ) = l4l + sl (0;) + 2dl Z 10 (0;0)0 d0;  0(2)and (; ) = r4r + sr (;) + 2dr Z 10 (; 0)0 d0;  0:(3)In the boundary conditions (2) and (3) i  0 for i = l; r. The coecients for specular(sl ; sr) and diuse (dl ; dr) re ection satisfydi ; si  0 and i + si + di = 1 for i = l; r.(4)The local albedo c(x) satises0  c(x)  1 for all x 2 [0;  ].Even though we consider the dimensionless form of the equations, we will stillrefer to  as temperature, as intensity, andf(x) = 12Z 0 (x; 0) d0(5)as the scalar ux. C. T. Kelley will present the paper.y North Carolina State University, Center for Research in ScienticComputationand Departmentof Mathematics, Box 8205, Raleigh, N. C. 27695-8205, USA. This researchwas supported by NationalScience Foundation grant #DMS-9321938 and a Cray Research Corporation Fellowship. Computingactivity was partially supported by an allocation of time from the North Carolina SupercomputingCenter. 1 The temperature  satises the diusion equation.@2@x2 = Q(x