The Efficient Iterative Solution of the $P1$ Equation

The $P1$ model is often used to obtain approximate solutions of the radiative transfer equation for heat transfer in a participating medium. For large problems, the algebraic equations used to obtain the $P1$ solution are solved by iteration, and the convergence rate can be very slow. This paper compares the performance of the corrective acceleration scheme of and Li and Modest (2002, “A Method to Accelerate Convergence and to Preserve Radiative Energy Balance in Solving the P1 Equation by Iterative Methods,” ASME J. Heat Transfer, 124, pp. 580–582), and the additive correction multigrid method, to that of the Gauss–Seidel solver alone. Additive correction multigrid is found to outperform the other solvers. Hence, multigrid is a superior solver for the $P1$ equation.