Precision Improvement of Numerical Simulation Including Iteration Processes for Nonlinear Evolution Equations

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For the purpose of construction of noble schemes in which influence of errors are extremely removed, we did consideration about setting of the reasonable condition for convergence of numerical solutions in iterative steps in solving the linear systems and about speed-up of convergence. In addition, we applied them to the issues of real fluid calculation and the problem of traffic jam in order to analyze several factors which govern nonlinear phenomena. First, we analyzed the structure of dynamical systems which are produced by discretizing the nonlinear differential equations from several viewpoints such as numerical experiments, visualization of nonlinear structure of attractors and statistical consideration. Concretely, we clarified essential structure of numerical solutions in discrete dynamical system and proposed the criterion to judge reliability of solutions.Next, we tried to discuss how structure of numerical solutions of a stochastic differential equation changes by the insertion of errors with the random style from the viewpoints of probabilistic approaches. Concretely, we tried to investigate the dependence of the structure of numerical solutions on insertion of random errors. As a fundamental study, the stochastic differential equation based on the deterministic logistic differential equation was considered and the relation between the size of noise and characteristics of obtained numerical solutions was discussed. Furthermore, we tried to discuss the dependence of the structure of numerical solutions of incompressible fluid equations on insertion of random errors in solving simultaneous equations.Next, direct numerical simulation to an integral equation of the first kind was carried out by using IPNS(Infinite-Precision Numerical Simulation). Numerical results are very satisfactory in accuracy. Moreover, they also show some interesting facts. These numerical results show IPNS facilitates numerical analysis for such inverse problems.