HEAT solves the heat equation using MPI.
The function H(X,T) is to be solved for on the unit interval 0.0 <= X <= 1.0 and the time interval 0 <= T <= 10.
The thermal conductivity function K multiplies the second derivative in the heat equation. We will be using an explicit method to solve our discretized version of the heat equation. In order that the results of such a computation are reliable, there is a relationship required between the time step, spatial step, and the function K. To avoid problems, we will need to set K = 0.002 / P, where P is the number of processors we are using. If we don't have a suitably small value of K, the program will refuse to carry out the computation.
The initial condition is H(X,T=0) = 95.0;
The boundary conditions are:
H(X=0,T) = 100 + 10 * sin ( T );
H(X=1,T) = 75.0
In order to divide the problem among P processors, the original domain is divided into P consecutive subintervals. Each processor is responsible for computing N values within its subinterval.
The discrete solution is computed for all spatial positions and a fixed time. We change notation, and now H(I) represents the I-th entry of a vector of values that are an estimate for the solution at one of these fixed times.
To get the next value of H(I), the processor needs current values of H(I-1), H(I) and H(I+1). This is fine except for H(1) and H(N), which are "missing" neighbors. In each case, the necessary information is available in the processor that is "next door".
To handle this situation, each processor actually sets up an array of N+2 values, including H(0) and H(N+1). When it comes time to update the values of H, the processor SENDS its values of H(1) to the left and H(N) to the right. It then RECEIVES values of H(0) from the left and H(N+1) from the right. Now that it has current values of H for entries 0 to N+1, it can compute new values for entries 1 to N.
Boundary conditions are used in the two special cases, that is, when process 0 needs H(0), and when process P-1 needs H(N+1).
Here is a diagram suggesting the situation when P = 3 and N = 4:
0 1 2 3 4 5 <== Nodes for process 0
: :
0 1 2 3 4 5 <== Nodes for process 1
: :
0 1 2 3 4 5 <== Nodes for process 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 <== Global node numbering
In this diagram, for instance, process 0 is really only responsible for computing new values of H at nodes 1, 2, 3 and 4. It keeps track of values at nodes 0 and 5 as well. It will get H(0) using a left boundary condition. And it will get H(5) when process 1 sends it the value of what process 1 calls H(1). In exchange, process 0 will be sending its value of H(4) to process 1.
The program starts with an initial condition at time T = 0, then takes 100 steps to reach time T = 10. There is data for a sequential program that used N = 12 interior points.
Each step of the iteration requires sending and receiving two messages, which are identified by message tags equal to 1 and to 2.
The code used for message tag #1 has been written for you already. The message with tag #1 is used when processors 0 through P-2 send H(N) to their neighbor on the right, processors 1 through P-1. Processors 1 through P-1 receive this number as H(0). Meanwhile, process 0 sets H(0) using a boundary condition.
You must finish the code used for message tag #2. The message with tag #2 is used when processors 1 through P-1 send H(1) to their neighbor on the left, processors 0 through P-2. Processors 0 through P-2 receive this number as H(N+1). Meanwhile, process P-1 sets H(N+1) using a boundary condition.
Your assignment is to take one of the source codes, make an MPI version, and compare the runs using 1 and 8 processes.
This assignment is to be turned in. You should turn in 3 files:
Before the due date, you may ask me for guidance or help.
The assignment is due by Friday, 26 September 2008.
Send your three files to Chris Harden.
Here are "completed" versions of the programs.
(Access to these files
will be restricted until everyone has turned in their work)
You can go up one level to the MPI page.
Last revised on 22 September 2008.