SOLUTION PROPERTIES OF THE VARIATIONAL PROBLEM FOR THE OPTIMIZATION OF POINT REDISTRIBUTION ON A SEQUENCE OF PROCESSING ELEMENTS

Yu.A. Bondarenko
VANT. Ser.: Mat. Mod. Fiz. Proc 1997. Вып.1. С. 46-47.

      For a sequence of processing elements (PE), we consider an explicit local algorithm similar to the explicit difference scheme with monotonic distribution of grid points m(x,t) on a PE where m is the point number, x is the PE number t is the timestep number. The algorithm is considered for the PE dynamic load balance accomplished by sending the grid points from one PE to the neighbors at each timestep. It is assumed that the computation time of the cell number m, σ(m,t), at the time t is a positive function continuously differentiated over m and α(m), the time of transfer of the, m cell from one PE to the neighbor, is also a positive function continuously differentiated over m. The functional
      
      is the computational time for a single timestep where m₀(x) = m(x, t - Δt) is the distribution of points over the sequence of PEs at the previous timestep. The time (1) should be minimized in terms of boundary conditions and monotonicity condition
      
      and monotonicity condition
      For this variational problem of step-by-step optimization of the computational time, the following theorems are proved.
      Theorem 1. If the functional S[m] reaches its minimum on the function m(x) satisfying the boundary conditions (2) and monotonicity condition (3) then this function satisfies equation
      
      Inversely, each continuous solution of equation (4) satisfying the boundary conditions (2) and monotonicity conditions (3) satisfies also the necessary condition of the minimum value of σS[m] > 0, Vσm(x).
      Theorem 2. Let the function m₀(x) be limited and continuously differentiated. Then equation (4) always has the solution {Φ, m(x)}, satisfying the boundary conditions (2), this solution is unique and the function m(x) is continuously differentiated.
      Theorem 3. Let the function m₀(x) be limited and continuously differentiated. If the function mo (x) satisfies the monotonicity condition and the boundary conditions
      m₀(x), _x=0 = 0, m₀(x), _x=X1 = M1, then the solution m(x) of equation (4) satisfying the boundary conditions (2) is also a monotonically growing continuously differentiated function.

      The report presented the results obtained with MIMGZA codes for the parallelization of 2-D gas-dynamics and heat conduction equations on the eight-processor MP-3 system. The computational domain is divided into subdomains in accordance with the number of processors. When the gas-dynamic equations are solved, the neighboring subdomains have the overlapping portions through which the processors communicate. The parallelization of 2-D heat conduction equation uses the pipeline algorithm arid the transposition algorithm. The results of two computations are given.