USE OF NORMA LANGUAGE FOR THE INTEGRATION OF POISSON EQUATION WITH VARIABLE COEFFICIENTS ON PARALLEL COMPUTERS

A.N. Andrianov, K.N. Efimkin, S.V. Zybin
VANT. Ser.: Mat. Mod. Fiz. Proc 1997. Вып.1. С. 39.

      The issues are considered for the use of declarative (nonprocedural) NORMA language to solve 2-D Poisson: equation in cylindrical coordinates on nonuniform grids in the problem of streamer propagation through the cathode layer.
      NORMA language is a tool designed for the specification of numerical methods for the calculations in computational physics on parallel computer systems. It actually allows to automate the programming phase needed to convert from the computational formulas specified by the applications specialists to the computer-specific code.
      There is no considerable difference between the computational formulas and NORMA, representation (of the algorithm; these formulas are the source information for the translation system. This description retains the natural parallelism of the problem without containing any restrictions related with the, wish to adapt the program to a parallel architecture or programming language features. NORMA representation does not require any information about the computational procedure, organization techniques for the computational (cyclic) processes. The order of language clauses can be arbitrary: the informational relations are identified and taken into account by the translator during the computational process organization. This resulted in the following high-level automatization of applications program development (the programmer operates primarily in terms of computational formulas from the application domain):
      — development of reliable applications programs (if the computational formulas are written, correctly, then the correct target program is guaranteed);
      — portability of NORMA programs (the synthesis NORMA translator takes into account the architecture features).
      In this paper NORMA language is used for the representation of the parallel.-algorithm for the calculation of 2-D Poisson equation in cylindrical coordinates using SOR method. The use of NQRMA allowed to obtain automatically the target Fortran program for IBM PG for a single (shared memory) node of the parallel Convex SPP-1000, and Fortran GNS program for. the distributed- memory i860XP parallel system with message passing. The source NORMA program was not actually changed; in the last case it is sufficient to add the specification of the number of processors to the, program that would be desirable for the execution., NORMA programming does not require the knowledge of complicated message passing mechanism; NORMA translator automatically organizes the computations and the communications between parallel tasks.
      The results obtained allow to suggest that the use of NORMA language for the calculations in computational, physics would considerably reduce the development cost of high-performance algorithms especially on parallel architectures.

      Parallel algorithms were considered for the integration of Cauchy problem for the system of ordinary differential equations implementing the large-scale time parallelism by using the iterative processes, particularly, the algorithms based on the discrete convolution.
      The parallelism of these algorithms rests on the transformation of the original ordinary differential equation to the equivalent integral equation and subsequent calculation through the iterations in time subintervals called blocks. The length of these subintervals is usually greater than the step of the grid used to calculate the integral which allows to achieve the large-scale parallelism where simultaneous calculations run in each iteration for the integrand at the grid nodes. These methods were not previously used on a single processor since they require O(n2) operations where n is the number of grid nodes. The emergence of parallel computers changed the situation, since only O(logn) operations are required on n processors for the iteration methods.
      The majority of iterative parallel algorithms also called “waveform relaxation methods” focus on block parallelization or when the integrand is calculated. This paper mainly considers the algorithms allowing additionally to use the potential parallelism of the discrete convolution. These algorithms rest on the approximate iterative Newton- Kantorovich method and are similar to the algorithms such as shifted Picard splitting. The basic idea is to solve linear ordinary differential equations at each iteration by evaluating the convolution integral. Upon the discretization with Gregory formulas the convolution integral can be calculated with parallel algorithms of the discrete convolution or using special processors (signal or pipeline).
      The theoretical results obtained include the studies of the convergence properties, approximation and stability of the algorithms used the estimates of the convergence rates of iteration processes used. The iterative processes were studied with the piecewise-constant Jacobi approximation (basic and modified) and the process with internal iterations. The structure was also developed for their parallel implementation and the family of parametrized algorithms is constructed with the hierarchy of the parallelism levels.
      These results allow to develop efficient parallel algorithms for the integration of ordinary differential equations based on Picard iterations (without convolution) and approximate Newton-Kantorovich iterations (with potential convolution use). The studies were performed for their computational properties on a set of different test ordinary differential equations the estimates are obtained for the acceleration and parallelization efficiency. The algorithms were also implemented as NORMA program to be translated to parallel Fortran GNS. The preliminary testing was accomplished for the generated program with the allocation of tasks over processors and support of communications on a parallel computer with i860XP processors.