Explicit and Implicit Taylor Series Based Computations

Differential equations, Taylor series method, TKSL, Stiff systems

This paper deals with computer simulations of continuous systems. The research group "High performance computing" has been working on extremely exact and fast solutions of homogenous differential equations, nonlinear ordinary and partial differential equations, stiff systems, large systems of algebraic equations, real time simulations and corresponding software and hardware (parallel) implementations since 1980. The Modern Taylor Series Method (MTSM) developed at our university is an original mathematical method which uses the Taylor series method for solving differential equations in a non-traditional way. Unfortunately, it is easier said than done as there are some peculiar systems of differential equations, which cannot be solved by commonly used (explicit) methods - the stiff systems.While the definition of this kind of systems is intuitively clear to the mathematicians the exact definition has not been specified yet. Often unnoticed, stiff systems showed up too often in practice such as in simulations of electrical circuits, chemical reactions and so on. To solve this kind of problems we can use for example multiple arithmetic or implicit numerical methods. In this paper we compare explicit and implicit Taylor series method in solution of the well-known Dahlquist's equation. We focus on stability and convergence of corresponding computations. Both explicit and implicit Taylor series methods give very successful results for the Dahlquist's equation and it can be expected that similar results will be obtained for sophisticated problems.