Parallel Computations of Differential Equations

Taylor series method, differential equations, initial values problems, parallel computations

The paper is focused on an original mathematical method which uses the Taylor series method for solving differential equations in a non-traditional way. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have shown and  theoretical analyses at the Department of Mathematics of the Faculty of Electrical Engineering and Communication of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. It has been verified that the computation quite naturally uses the full hardware accuracy of the computer and is not restricted to the usual accuracies of $10^{-5}$ to $10^{-6}$. It has also been verified that the computation speed enabled by the newly developed Taylor series method is, while keeping the high accuracy, greater than that achieved by the algorithms currently used for numerically solving systems of differential equations. This feature is accentuated especially while solving large scale systems of linear differential equations. The Modern Taylor Series is based on a recurrent calculation of the Taylor series terms for each time interval. Thus the complicated calculation of higher order derivatives (much criticised in the literature) need not be performed but rather the value of each Taylor series term is numerically calculated. Solving the convolution operations is another typical algorithm used. An important part of the method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires. Thus it is usual that the computation uses different numbers of Taylor series terms for different steps of constant length. An automatic transformation of the original problem is a necessary part of the Modern Taylor Series Method. The original system of differential equations is automatically transformed to a polynomial form, i.e. to a form suitable for easily calculating the Taylor series forms using recurrent formulae. The "Modern Taylor Series Method" also has some properties very favourable for parallel processing. Many calculation operations are independent making it possible to perform the calculations independently using separate processors of parallel computing systems. Since the calculations of the transformed system (after the automatic transformation of the initial problem) use only the basic mathematical operations (+,-,*,/), simple specialised elementary processors can be designed for their implementation thus creating an efficient parallel computing system with a relatively simple architecture (first experiments have been done using the Xilinx FPGA gate array).