Extremely Accurate Solutions of Systems of Differential Equations
The development project deals with extremely exact, stable and fast numerical solutions of systems of differential equations. In a natural way, it also involves solutions of problems that can be reduced to solving a system of differential equations.
The project is based 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. On the other hand, for a pre-set integration order, the integration step length may be selected. This fact positively affects the stability and speed of the computation.
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 Seriers 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. This parallel approach has been tested using the available parallel transputer system - an original methodology for parallel computation of systems of differential equations in a transputer system has been defined.
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).
The work includes experimental computations performed using a tailor made multiple arithmetic as an important part. It is characteristic of the Modern Taylor Series Method that the computation accuracy for a given step length is increased with the number of Taylor series terms used. However this increase in accuracy is not unlimited. For a given integration step length, there is always a saturated computation error that depends on the arithmetic unit word length. In some cases this saturated computation error can be reduced by decreasing the integration step or increasing the arithmetic unit word length. The effect of increasing the arithmetic unit word length is more significant than that of reducing the integration step length.