Parallel Computing with MPI - Part II: Discretization of the Heat Equation

In this article, we'll discuss how to discretize the heat equation using an Euler explicit differencing schem. The discretization is first order forward in time, and second order central in space. I am assuming that you are somehow familiar with finite difference methods. If not, then discuss it with your therapist.

Again, we are considering the heat equation given by

The underlying idea in discretization is to first divide the domain of interest (i.e. the region over which we are solving the heat equation) into a grid of discrete points (in space and time). When this is done, the derivatives can be approximated using the values of the dependent variable T at the discrete points. For convenience, the grid points are indexed using i in the horizontal x direction, and j in the vertical y direction.

Similarly, the time dimension is divided into time levels n, n+1, n+2... separated by a timestep Δt.

Next, we have to transform Eq. (1) into an algebraic relation at (any) point (i, j) and time level n. To do this, we have to convert each derivative into an equivalent algebraic expression. For the unsteady term, we use a forward Euler scheme as follows


where n denotes the time level and Δt denotes the timestep size. The form given in Eq. (2) is called forward in time because it uses the value of T at the next time level (n+1), which is unknown. In essence, we are solving for T at (n+1)

For the spatial derivatives, we will use explicit central differencing. This means that the discretized form of the spatial derivatives will be evaluated at the current time level n where the values are known. This is written as

and
If we further allow Δx = Δy = δ Upon substitution of Eqs. (2)-(4) into Eq. (1), we obtain the full discretized form of the heat equation


finally, the simplicity of the explicit scheme is clearly visible in Eq. (5) as one can easily solve for the new time level (n+1)
This means that there is no need to solve a linear system of equations. If we had used an implicit method, the RHS of Eq. (6) would be at time level (n + 1) and therefore, one would have to solve a linear system of equations at each time step.

In the next post, we'll discuss the stability of the discretized equation and derive the CFL criterion to ensure stability of the numerical solution given by Eq. (6).

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 II: Discretization of the Heat Equation". Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/06/parallel-computing-with-mpi-part-ii.html