Parallel Computing with MPI - Part I: The Heat Equation

In this series, I will introduce parallel programming concepts with MPI. This series is prompted by the "Numerical Methods for PDEs" class given by Professor Christian Parigger of UTSI.

Of course, since things are done best with concrete examples; our objective will be to parallelize the 2D heat equation (i.e. unsteady diffusion) with constant properties. The parallelization of the 2D heat equation is actually one of the problems on the final exam of Prof. Parigger's class and is based on the paper by Horak and Gruber, Parallel Numerical Solution of 2-D Heat Equation.

Here's an outline of what I am planning to talk about:

1. Introducing the Heat Equation
2. Discretization of the Heat Equation
3. Stability Analysis and the Courant-Friedrichs-Lewy Criterion
4. Programming the Heat Equation in C using Visual Studio
5. MPICH2 and Visual Studio
6. The Essential Ingredients of an MPI Program
7. MPI Warmup
8. Parallelizing the Heat Equation (2 or 3 posts - most likely a video tutorial)

The heat equation is a great example to introduce parallel computing because we will use an explicit scheme for the discretization. The reason for the simplicity of the explicit schemes and its suitability for parallel computing will become clear as I progress in this series. (For those who absolutely want to know why it is simpler to parallelize an explicit finite difference scheme, I'll briefly point out that it is because you are not solving a linear system of equations. When using explicit methods, the values at the new time step are explicitly expressed in terms of the values at the previous one and therefore the exchange of values at the interface is straightforward).

So let us first start by introducing the heat equation.

The heat equation (with no heat sources) expresses a balance between the accumulation and the flow of heat within a certain region of space, usually referred to as a control volume. Obviously, the heat equation is not only restricted to heat. Many other physical phenomena share the same form of the heat equation such as mass transfer, momentum transfer, and Shroedinger's equation. In its simplest forms, the heat equation describing heat diffusion in solids (aka conduction heat transfer) is given by

ρ is the density of the material, c is the heat capacity (if we had a fluid, c would be the heat capacity at constant pressure or cp). It describes how much heat a solid (or a fluid element) can store and subsequently raise the temperature as opposed to letting it pass through. k is the thermal conductivity of the solid and describes the resistance to the flow of heat. A higher k means more heat can flow through the material. For example, insulators have a very low thermal conductivity while heat sinks (such as CPU heat sinks) have very large k.

The unsteady term at the LHS of Eq. (1) describes how much heat is accumulated in a control volume as time goes by. On the other hand, the RHS describes how much heat is flowing in the spatial directions.

Fundamentally, the heat equation describes a process called diffusion, which denotes the transfer of "information" (e.g. heat) at the molecular level. This is useful when convection enters the picture because it occurs at the macroscopic level, in the presence of a velocity field associated with fluid flow (I'll explain this more in a future post).

Mathematically, diffusion is almost always described by second derivatives in space (while convection is associated with first derivatives in space). This is clearly visible in Eq. (1) above. A more convenient way of writing the heat equation is by dividing both sides of Eq. (1) by ρc. The outcome isThe new parameter, α, shown in Eq. (2) is called the thermal diffusivity. It is a measure of how much a solid can pass heat or store it. For example, large values of alpha mean that heat can flow through material very fast, while small values indicate a tendency for the material to store heat.

In the next post, we will discretize Eq. (2) using explicit forward differencing in time, and central differencing in space.

Table of Contents:

1. Introducing the Heat Equation
2. Discretization of the Heat Equation
3. Stability Analysis and the Courant-Friedrichs-Lewy Criterion
4. Programming the Heat Equation in C using Visual Studio
5. MPICH2 and Visual Studio
6. Parallel Computing in a Nutshell
7. Measuring Parallel Performance
8. Essential Ingredients of Message Passing
   
9. MPI Warmup
   
10. Parallelizing the Heat Equation

Cite as:
Saad, T. "Parallel Computing with MPI - Part I: The Heat Equation". Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/06/parallel-computing-with-mpi-part-i-heat.html