Augmented Quadratures for the Discrete Ordinates Method using Reduced Order Modeling Approaches

John Tencer, Kevin Carlberg, Marv Larsen, and Roy Hogan
Pacific Rim Thermal Engineering Conference 13–17 March 2016 Waikoloa Village, HI, USA
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Abstract

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat transfer applications, a quasi-steady assumption is valid. The dependence on wavelength is often treated through a weighted sum of gray gases type approach. The discrete ordinates method is the most common method for approximating the angular dependence. In the discrete ordinates method, the intensity is solved exactly for a finite number of discrete directions. Integrals over the angular space are accomplished through a quadrature rule. In this work, a projection-based model reduction approach is applied to the discrete ordinates method. A low-order quadrature is used to establish the reduced basis. The reduced model is then queried to effectively increase the quadrature order. This results in a much more accurate solution than was achieved by the low-order quadrature. One-and two-dimensional test problems are presented.