Abstract. The article presents a method of computing the step response of an air floor heater. The method implements parallel algorithms on a graphics processing unit. In the analyzed concrete slab heating ducts are placed. Hot air is transferred through them, thanks to which the heat penetrates into the slab. Heat transfer into the environment takes place on the top surface of the floor by natural convection and radiation. The bottom surface of the slab is thermally insulated. A two-dimensional heat equation was discretized with the use of the implicit finite difference method. In order to solve the obtained system of equations, the conjugate gradient method was used. Moreover, in order to examine the possibility of shortening the computations time, the algorithm of this method was implemented on a graphics processing unit. A computer program, using the CUDA parallel computing platform and linear algebra libraries CUBLAS and CUSPARSE, was developed.

Purpose -To present a new parallel method for solving differential equations that describe transient states in physical systems. Design/methodology/approach -The proposed speculative method first solves a differential equation with a large integration step to determine initial data for parallel computations in sub-intervals of time, then speculatively computes in parallel solutions in all the sub-intervals with a smaller integration step and finally composes the final solution from the speculatively computed ones. The basic numerical method applied is the well-known Runge-Kutta algorithm. Findings -The speculative method allows important reduction of the computation time of sequential algorithms. The speed-up of the speculative method that we propose, as compared to the sequential execution, depends on the number of sub-intervals that are defined inside the total analysed time interval. The speed-up increases almost linearly with the number of sub-intervals. The good accuracy of computations in the presented example was obtained.Research limitations/implications -The proposed method can be applied to non-linear systems without discontinuity points and to stable systems (i.e. systems insensitive to the selection of initial conditions). Practical implications -The method can be especially applied for long-lasting computations with a slow convergence of state variables values along with the decrease of integration steps. Originality/value -The paper presents an original parallel method for solving differential equations, which significantly speeds up transient states analysis in physical systems.

In this paper the parallel method for the transient state analysis in electrical circuits, described by a system of ordinary differential equations, is presented. The general idea of this method is based on decomposition of the analyzed time domain into sub-intervals. Parallel computations in particular sub-intervals of time are conducted with the initial conditions determined on the basis of an approximation of convergent graph solution by exponential function and with the use of sequential numerical Runge-Kutta method.

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