Analysis of a fractional-step scheme for the P1 radiative diffusion model
Provides rigorous numerical analysis for a challenging physical model, but the scheme is incremental (finite volume with fractional steps) and the results are theoretical rather than empirical.
The paper proposes a finite volume fractional-step scheme for a nonlinear parabolic system modeling P1 radiative diffusion, proving existence, uniqueness, and convergence of discrete solutions to the continuous problem via compactness arguments.
We address in this paper a nonlinear parabolic system, which is built to retain the main mathematical difficulties of the P1 radiative diffusion physical model. We propose a finite volume fractional-step scheme for this problem enjoying the following properties. First, we show that each discrete solution satisfies a priori L -estimates, through a discrete maxi- mum principle; by a topological degree argument, this yields the existence of a solution, which is proven to be unique. Second, we establish uniform (with respect to the size of the meshes and the time step) L2 -bounds for the space and time translates; this proves, by the Kolmogorov theorem, the relative compactness of any sequence of solutions obtained through a sequence of discretizations the time and space steps of which tend to zero; the limits of converging subsequences are then shown to be a solution to the continuous problem. Estimates of time translates of the discrete solutions are obtained through the formalization of a generic argument, interesting for its own sake.