On the Smoothness of the Solution to the Two-Dimensional Radiation Transfer Equation

TL;DR

This paper investigates the smoothness properties of solutions to the two-dimensional radiation transfer equation, providing derivative estimates and demonstrating how solution regularity affects numerical method convergence.

Contribution

It offers new derivative estimates for the scalar flux solution and links solution smoothness to the convergence of the diamond difference numerical method.

Findings

Derivative estimates near the boundary are established.

Solution smoothness influences the convergence rate of numerical schemes.

Numerical example confirms theoretical implications.

Abstract

In this paper, we deal with the differential properties of the scalar flux defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation. Estimates for the derivatives of the scalar flux near the boundary of the domain are given based on Vainikko's regularity theorem. A numerical example is presented to demonstrate the implication of the solution smoothness on the convergence behavior of the diamond difference method.