Introductory Finite Volume Methods for PDEs

This material is taught in the BSc. Mathematics degree programme at the Manchester Metropolitan University, UK. The Finite Volume Method (FVM) is taught after the Finite Difference Method (FDM) where important concepts such as convergence, consistency and stability are presented. The FDM material is contained in the online textbook, ‘Introductory Finite Difference Methods for PDEs’ which is free to download from this website

It is recommended that the FDM text book is read before this book. This textbook is also freely downloadable from the above website.

The following chapters contain core material supported by pen and paper exercises together with computer-based exercises where appropriate. Supporting material, including worked solutions, and computer codes can be found at the companion website:

www2.docm.mmu.ac.uk/STAFF/C.Mingham

Codes, with which the student can experiment, are written using Matlab. In the spirit of Open Source, it is hoped to reproduce these codes using Scilab (a Matlab clone, downloadable for free from www.scilab.org).

The emphasis of this book is on a practical understanding of the basics of the FVM and a minimum of theory is given to underpin the method. Revision material is provided in appendices. It is recommended that anyone wishing to use the FVM to solve systems of equations for real world applications reads up on the underlying physics of the problem.

This book is intended for final year undergraduates who have knowledge of Calculus, vectors and introductory level computer programming.

1. Introduction
   1. Introduction
   2. Obtaining the Integral Form from the Differential Form
   3. Finite Volume Meshes
   4. Discretising the Semi-Integral Equation
   5. Implementation
2. Finite Volume Schemes
   1. Introduction
   2. FVM on a Cartesian Mesh
   3. Finite Volume Schemes in 1D and 3D
   4. Time Step Calculation for a Finite Volume Scheme
   5. Finite Volume FOU 2D Scheme
   6. Boundary Conditions
   7. Coding a Finite Volume Solver
3. Derivation of Equations
   1. Introduction
   2. Conservation Laws
   3. Control Volume Approach
   4. Deriving the Integral Form of the 2D Linear Advection Equation
   5. Deriving the Differential Form of the 2D Linear Advection Equation
4. Further Finite Volume Schemes
   1. Introduction
   2. Linear Interpolation
   3. Quadratic Interpolation
   4. Converting from Finite Difference to Finite Volume
5. Systems of Equations
   1. Introduction
   2. The Shallow Water Equations
   3. General FVS for the SWE
   4. FVS for the 2D SWE on a Structured Mesh
   5. Heuristic Time Step for a 2D SWE FVS