STABILITY OF DIFFERENCE SCHEMES FOR PARALLELIZATION IN TERMS OF PHYSICAL PROCESSES

Yu.A. Bondarenko
VANT. Ser.: Mat. Mod. Fiz. Proc 1994. Вып.2. С. 3-5.

      The results of stability studies are presented for difference schemes, in which different processes are calculated independently at the same time step on different processors. Then computation results are summed up and used for the next step calculation. In a large multi-parameter family of one-step difference schemes for parallel calculation of 2 processes one did not find difference schemes with 2nd order of accuracy over time and absolutely stable at the same time. The simplest implicit difference scheme for parallel calculation of 3 and more processes with 1st order of accuracy unlike the analogical splitting scheme turned out to be conditionally stable. The family of absolutely stable schemes is constructed for this case.

      Some three-layer difference schemes, explicit and implicit ones, are constructed for linear equation d²u/dt²+ A(t)u = 0 using the method of discrete approximation for operation functional by Gamilton-Ostrogradskiy with the 4^th order accuracy. The variational difference schemes constructed approximate initial equation with only the 2^nd order accuracy, when operator A(t) is a function of time t. It appears that there always exists a substitution of difference solution of the form u(tⁿ) = (1 + τ²B(tⁿ))v(tⁿ) so that for a new unknown function a difference scheme approximates initial equation with the 4^th order accuracy, that is the real accuracy has the 4^th order accuracy and corresponds to the error of the operation functional approximation, but it is latent. The whole analysis is performed for a constant time step.