# Time-dependent simplified Pn equations in two dimensions

Authors
Publisher
Technische Universiteit Eindhoven
Publication Date
Source
Legacy
Disciplines
• Mathematics
• Medicine

## Abstract

Time-Dependent Simplified PN Equations in Two Dimensions Yabin Fan Supervisor: Prof. Dr. Axel Klar, Dr. Martin Frank November 29, 2007 Abstract Radiotherapy has been used in medicine for more than 100 years, the radiative transport equation plays a very important role in radiotherapy mathematically. The steady-state simplified PN approximation to the radiative transport equation has been successfully applied to many problems involving radiation. In this work, we use asymptotic analysis to derive the time-dependent SPN equations. And also three numerical test cases are given, including Marshak wave and Lattice problem. Contents 1 Introduction 3 2 SPN equations 5 2.1 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The steady-state SPN equations . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Time-dependent SPN equations via asymptotic analysis . . . . . . . . . . . 9 2.3.1 SP1 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 SP3 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.3 SSP3 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Boundary and initial conditions 14 3.1 Boundary condition for SP1 . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Boundary conditions for SP3 . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Numerical method 19 4.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Test case 22 5.1 Marshak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.1.2 SPN equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.1.3 Numerical results . . . . . . . . . . . . . . . . . .

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