CS 575 Supercomputing - Lab 4: Problem Solving Process
February 21, 2001

Dr. Kris Stewart (stewart@sdsu.edu)
San Diego State University

1. Subdivide the rectangular plate
   • pick m (number of nodes in a row)
   • pick k (number of rows
   • number of unknowns N = m * k
   • write out the N equations that represent heat diffusion where is is known that the temperate at an interior node is the average of the temperate at neighbor nodes

                                   x(j+m)
                                     |
                         x(j-1) --- x(j) ---- x(j+1)
                                     |
                                   x(j-m)
     

     The general case is then: 4 * x(j) = x(j-1) + x(j+1) + x(j-m) + x(j+m) when we examine nodes well in the interior of our grid. The special nodes in the corners and the edges have their own equations, for example,
     • for j=1
       4 * x(1) = 20 + x(2) + 10 + x(m+1)
       or separating knowns and unknowns
       4 * x(1) - x(2) - x(m+1) = 30
     • for j=2
       4 * x(2) = x(1) + x(3) + 20 + x(m+2)
       or 4 * x(2) - x(1) - x(3) - x(m+2) = 20
     • I wouuld suggest sitting down with a large piece of paper,
       • selecting specific values, say m=10 and k=6
       • draw the grid for the N = m*k = 10*6 =60 unknowns
       • write out the 60 equations (or at least enough until you see the pattern
       • assemble in the form A x = b, i.e. put unknowns on one side of the equations and knowns (b) on the other side of the equal mark
2. Assemble the matrix A (as outlined above)
3. Assemble the right hand side, b, containing the known values
4. Solve the linear system A x = b using Gaussian Elimination with partial pivoting
   • Factor the matrix A = L U, where L is a lower triangular matrix and U is an upper triangular matrix.
   • Solve the system A x = L U x = b