The Concurrent wave Equation

It is described in Chapter 5 of Fox et al. (1988), Solving Problems on Concurrent Processors, Vol 1. The implementation needs to calculate the amplitude of points along a vibrating string for a specified number of time intervals. The equation to solve is:

"i" indicates the point on the line. The values array holds the current amplitudes. Note that the new amplitude for the point will depend on the current values at neighboring points.

The decomposition can be viewed as:

• Each process is assigned a contiguous block of points ("block" decomposition).

• Each process has all the data needed to update its interior points.

• To update its endpoints, the process must receive values for the points bordering the block (boundary values) from the processes that "own" them.

• It must also send the values of its own endpoints to these tasks. Non-blocking sends and receives are used for this communication, followed by a call to MPI_Wait. Thus, the main computational loop exchanges endpoints and updates amplitudes. (see version 1 of the implementation).

• Since the endpoint exchanges in the main computational loop use the same arguments for the message-passing calls for all the loop iterations, and the loop is repeated many times, they are good candidates for persistent communication (see version 3 of the implementation).

• Version 2 uses "shift" in the update step. This is done via Sendrecv to the left and right in a non-circular fashion. The rest of the program is similar to version 1.

• In these implementations, the master process reads the initial amplitudes from a file or the command line. Blocks of data must be distributed to workers. Similarly, at the end of the run, the updated data must be collected on the master for writing to a file.

• Using persistent communication, the program steps are:

  begin program
  
     learn number of processes and rank 
  
     identify left and right neighbors 
  
     get program parameters 
  
     initialize wave values 
  
     create persistent communication  
  
     update values  
  
     deallocate persistent communication  
  
     if master: collect results and print 
  
     if worker/slave: send results to master 
  end
  

• For the Send/Rcev version, the steps roughly as follows:

  begin program
  
     learn number of processes and rank
  
     if master: get program parameters and initial data
  
     broadcast program parameters
  
     distribute initial data  
  
     update values
  
     collect results  
  
     if master: store results
  
  end
  

• The code prints out several values for validation:

  
   1: .00   11:-.20   21:-.32   31:-.32   41:-.19   51: .01   61: .21   
  71: .32   81: .31   91: .18  100: .00