Introductory Finite Volume Methods for PDEs

Cover......Page 1
Title Page......Page 2
Copyright page......Page 3
Contents......Page 4
Preface......Page 7
1.2 Obtaining the Integral Form from the Differential Form......Page 8
1.3 Finite Volume Meshes......Page 10
1.4 Discretising the Semi-Integral Equation......Page 12
1.5 Implementation......Page 14
2.1 Introduction......Page 18
2.2 FVM on a Cartesian Mesh......Page 19
2.3 Finite Volume Schemes in 1D and 3D......Page 22
2.4 Time Step Calculation for a Finite Volume Scheme......Page 25
2.6 Boundary Conditions......Page 27
2.7 Coding a Finite Volume Solver......Page 29
3.3 Control Volume Approach......Page 33
3.4 Deriving the Integral Form of the 2D Linear Advection Equation......Page 35
3.5 Deriving the Differential Form of the 2D Linear Advection Equation......Page 36
4.2 Linear Interpolation......Page 39
4.3 Quadratic Interpolation......Page 42
4.4 Converting from Finite Difference to Finite Volume......Page 43
5.1 Introduction......Page 48
5.2 The Shallow Water Equations......Page 47
5.3 General FVS for the SWE......Page 49
5.4 FVS for the 2D SWE on a Structured Mesh......Page 51
5.5 Heuristic Time Step for a 2D SWE FVS......Page 52
A.2 Review of Vectors......Page 55
A.3 Side Vectors......Page 57
B.2 Vector Calculus......Page 60
B.3 Line Integrals and Double Integrals......Page 61
B.4 Green’s Theorem......Page 63
B.5 Approximation of the Line Integral......Page 65
C.1 Introduction......Page 68
C.3 Code Listing......Page 69
C.4 Commentary......Page 79