A Look at Transport Theory from the Point of View of Linear Algebra
Vance Faber, Thomas A. Manteuffel · 2023
We show that the notion of “preconditioning” from linear algebra can provide a framework for the discussion of algorithms for the numerical solution of the transport equation. In this context we show that the conjugate gradient method yields substantial savings over the Neumann series solution of the standard integral formulation of the transport equation for optically thick regimes. Further, we show that the diffusion synthetic acceleration (DSA) algorithm is the Neumann series solution of the standard integral formulation of the transport equation preconditioned by the Green’s function of a diffusion operator. The preconditioned conjugate gradient method, using DSA as a preconditioner, again yields substantial savings.