|Budget Amount *help
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1995 : ¥400,000 (Direct Cost : ¥400,000)
Fiscal Year 1994 : ¥1,600,000 (Direct Cost : ¥1,600,000)
Head investigator, in the close coordination with SAITO and other people, has carried out the research work of the above and other relevant topics of numerical analysis. The main concern is on the mathematical modelling through differential equations as well as on the numerical solution of such equations. They include large system of ordinary differential equations (ODE_S), stochastic differential equations (SDE_S) as a stochastic extension of ODE_S, delay-differential equations (DDE_S) and others. The project has been mainly devoted in the stability and analysis of their numerical algorithms. The achievement follows.
(a) As for the numerical solution of SDE_S, we can establish the algebraic structure of ROW-type schemes, especially their rooted tree analysis, the issues of their implementation on computer, arrangement of the concepts of numerical stability for SDE_S, and criteria of M- and T-stabilities which were proposed by ourselves.
(b) We studied the parallel implementation of the
implicit Runge-Kutta methods for ODE_S. In particular, we proposed the collocation-type two-step Runge-Kutta methods as a parallelization of them. Some initial performance tests were also carried out for them.
(c) We studied the numerical as well as analytical stability of DDE_S, and established criteria of absolute stability for linear multistep and Runge-Kutta methods when applied to DDE_S.
(d) Studying the iterative solution of large system of linear algebraic equations with sparse coefficient matrices, we pointed out features of the convergence behavior of a series of the product-type iterative methods derived from the conjugate gradient method.
The topics described above are strongly expected to attract further interests in both theoretical and practical point of view. However, our achievement remains some questions, in particular in their implementation issues. Henceforth we hope a new research project succeeds ours to obtain more fruitful achievement.