# Analysis of a fractional-step scheme for the P1 radiative diffusion model

## TL;DR

This paper develops and analyzes a finite volume fractional-step scheme for the P1 radiative diffusion model, proving existence, uniqueness, and convergence of solutions with uniform bounds.

## Contribution

It introduces a novel fractional-step scheme for the nonlinear radiative diffusion model and provides rigorous proofs of stability, existence, uniqueness, and convergence of the discrete solutions.

## Key findings

- Discrete solutions satisfy a priori L-estimates
- Existence and uniqueness of solutions are established
- Convergence to the continuous problem is proven

## Abstract

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.