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Scattering And Absorption of Light in Planetary Regoliths

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Published on: July 1, 2019

Modelling with measures: approximation of a mass-emitting object by a point source.

Joep H M Evers¹, Sander C Hille, Adrian Muntean

¹Institute for Complex Molecular Systems and Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands. j.h.m.evers@tue.nl.

Mathematical Biosciences and Engineering : MBE

|March 27, 2015

Summary

This study approximates diffusion dynamics in a bounded domain using a point source model. It provides bounds on the approximation error for flux and solution differences.

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Area of Science:

Mathematical modeling
Partial differential equations
Numerical analysis

Background:

Diffusion equations model physical processes like heat transfer and fluid flow.
Accurate approximation of boundary conditions is crucial for realistic simulations.
Modeling complex geometries can be computationally intensive.

Purpose of the Study:

To develop and analyze an approximation method for linear diffusion equations on bounded domains.
To introduce a simplified model using a measure-valued point source in R(2).
To quantify the approximation error for both boundary flux and solution values.

Main Methods:

Solving a linear diffusion equation on a bounded domain with prescribed boundary flux.
Approximating the solution using a diffusion equation on the entire R(2) with a point source.
Deriving L(2) bounds for the difference in boundary flux and solution between the two models.

Main Results:

An L(2)([0,t];L2(Γ))-bound was derived for the boundary flux difference.
An L(2)(Ω)-bound and an L2([0,t];H(1)(Ω))-bound were established for the solution difference.
These bounds quantify the accuracy of the point source approximation.

Conclusions:

The point source approximation provides a valid method for simplifying diffusion problems on bounded domains.
The derived error bounds are essential for assessing the reliability of this approximation technique.
This approach offers a computationally efficient alternative for certain diffusion modeling scenarios.