Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods

Abstract

The ability to solve the radiative transfer equation in a fast and accurate fashion is central to several important applications in combustion physics, controlled thermonuclear fusion and astrophysics. Most practitioners see the value of using discrete ordinates methods for such applications. However, previous efforts at designing discrete ordinates methods that are both fast and accurate have met with limited success. This is especially so when parts of the application satisfy the free streaming limit in which case most solution strategies become unacceptably diffusive or when parts of the application have high absorption or scattering opacities in which case most solution strategies converge poorly. Designing a single solution strategy that retains second-order accuracy and converges with optimal efficiency in the free streaming limit as well as the optically thick limit is a challenge. Recent results also indicate that schemes that are less than second-order accurate will not retrieve the radiation diffusion limit. In this paper we analyze several of the challenges involved in doing multidimensional numerical radiative transfer. It is realized that genuinely multidimensional discretizations of the radiative transfer equation that are second-order accurate exist. Because such discretizations are more faithful to the physics of the problem they help minimize the diffusion in the free streaming limit. Because they have a more compact stencil, they have superior convergence properties. The ability of the absorption and scattering terms to couple strongly to the advection terms is examined. Based on that we find that operator splitting of the scattering and advection terms damages the convergence in several situations. Newton–Krylov methods are shown to provide a natural way to incorporate the effects of nonlinearity as well as strong coupling in a way that avoids operator splitting. Used by themselves, Newton–Krylov methods converge slowly. However, when the Newton–Krylov methods are used as smoothers within a full approximation scheme multigrid method, the convergence is vastly improved. The combination of a genuinely multidimensional, nonlinearly positive scheme that uses Full Approximation Scheme multigrid in conjunction with the Newton–Krylov method is shown to result in a discrete ordinates method for radiative transfer that is highly accurate and converges very rapidly in all circumstances. Several convergence studies are carried out which show that the resultant method has excellent convergence properties. Moreover, this excellent convergence is retained in the free streaming limit as well as in the limit of high optical depth. The presence of strong scattering terms does not slow down the convergence rate for our method. In fact it is shown that without operator splitting, the presence of a strong scattering opacity enhances the convergence rate in quite the same way that the convergence is enhanced when a high absorption opacity is present! We show that the use of differentiable limiters results in substantial improvement in the convergence rate of the method. By carrying out an accuracy analysis on meshes with increasing resolution it is further shown that the accuracy that one obtains seems rather close to the designed second-order accuracy and does not depend on the specific choice of limiter. The methods for multidimensional radiative transfer that are presented here should improve the accuracy of several radiative transfer calculations while at the same time improving their convergence properties. Because the methods presented here are similar to those used for simulating neutron transport problems and problems involving rarefied gases, those fields should also see improvements in their numerical capabilities by assimilation of the methods presented here.