1.0 Introduction
This is an individual coursework. The purpose of this coursework is to investigate the effect that the choice of numerical schemes for convection terms has on the solution of numerical problems using the Finite Volume Method (FVM). A secondary objective is to implement solutions to a simple 1D convectiondiffusion problems using computational tools. Essentially, you will be programming a simple CFD code. Finally, the entire process will be critically discussed in writing and it will be presented via a formal technical report. The procedure to complete this coursework is summarised below and further explained in this brief.
 To develop a discretised solution for the 1D diffusionconvection equation
 To develop a numerical implementation of the diffusionconvection equation using the central difference scheme and the Finite Volume Method (FVM)
 To carry out a literature investigation to identify the general conditions and requirements for numerical schemes. From this investigation an upwindbiased scheme will be identified for implementation.
 To modify the initial numerical implementation (2) to incorporate an upwindbiased scheme (identified in step 3 above) for convective terms only (central difference should be applied to diffusion terms)
 To critically discuss and present findings via a formal technical report.
2.0 Discretised solution of the convectivediffusion equation using the using the Central Difference scheme and the Finite Volume Method
The first task involved in this course is the development of a general analytical solution for the model 1D convectiondiffusion equation:
π π ππ
(πππ) = (Γ )
ππ₯ ππ₯ ππ₯
And the flow must also satisfy the 1D continuity equation
π
(ππ) = 0
ππ₯
For the development of the discretised general solution, the flow can be assumed to be incompressible, and the velocity field known. The discretised solution must be developed within the Finite Volume Method framework using the Central Difference scheme on all terms. Students are expected to use the classic notation P, E, W, e and w to denote the point of interest, the cellcentre point of the cell to the East, the cellcentre point of the cell to the West, and the east and west cellface values, respectively.
3.0 Development of a numerical solution and its implementation using the Central Difference scheme and the Finite Volume Method
Once a general discretised solution for the diffusionconvection equation in 1D has been developed. A numerical algorithm to solve the equation (preferably using MatLab or Octave, however, other programming languages will also be accepted) will be developed using the Finite Volume Method and the Central Difference Scheme (for all equation terms). This code will be developed to solve the following 1D convectiondiffusion problem:
