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Parallel numeric solution of differential equations
Nečasová, Gabriela ; Čermák, Martin (oponent) ; Kozek, Martin (oponent) ; Šátek, Václav (vedoucí práce)
Differential equations have been studied for over 300 years. Partial differential equations were first used by the Swiss mathematician and lawyer Nicolaus Bernoulli in the 18th century. Second-order partial differential equations are used to model a wide range of phenomena in science, engineering, and mathematics, such as the propagation of light and sound waves, the motion of fluids, and the diffusion of heat. The thesis deals with the parallel numerical solution of partial differential equations. Second-order partial differential equations are transformed into large systems of ordinary differential equations using the method of lines. The spatial derivatives in the partial differential equation are replaced by various types of finite differences. The resulting large systems of ordinary differential equations (initial value problem) are solved in parallel using Runge-Kutta methods and the newly proposed higher-order method based on Taylor series. The numerical experiments of the selected problems are calculated using a supercomputer with different numbers of compute nodes. The results show that the Taylor-series-based numerical method significantly over-performs state-of-the-art Runge-Kutta methods.
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Advanced Electronic Circuits Simulation Methods
Kocina, Filip ; Kozek, Martin (oponent) ; Kyncl, Jan (oponent) ; Kunovský, Jiří (vedoucí práce)
The thesis deals with the simulation of electronic circuits. It describes the Capacitor Substitution Method (CSM) to transform electronic circuits into electric circuits which can then be solved using numerical methods, namely the Modern Taylor Series Method (MTSM). This method is distinguished by automatic order selection, halving the step size as required and the wide area of stability according to the order. Within the thesis, specialized programming equipment to solve ordinary differential equations using MTSM was created by the author of the thesis, with many improvements to the algorithms (compared to TKSL/386). These algorithms involve the simplification of generic expressions into polynomials, parallelization independent of the integration method etc. This software runs on a Linux server which communicates using the TCP/IP stack. The equipment was successfully used to simulate VLSI circuits whose solution by CSM was much faster and more memory-efficient than the state-of-the-art SPICE.
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Advanced Electronic Circuits Simulation Methods
Kocina, Filip ; Kozek, Martin (oponent) ; Kyncl, Jan (oponent) ; Kunovský, Jiří (vedoucí práce)
The thesis deals with the simulation of electronic circuits. It describes the Capacitor Substitution Method (CSM) to transform electronic circuits into electric circuits which can then be solved using numerical methods, namely the Modern Taylor Series Method (MTSM). This method is distinguished by automatic order selection, halving the step size as required and the wide area of stability according to the order. Within the thesis, specialized programming equipment to solve ordinary differential equations using MTSM was created by the author of the thesis, with many improvements to the algorithms (compared to TKSL/386). These algorithms involve the simplification of generic expressions into polynomials, parallelization independent of the integration method etc. This software runs on a Linux server which communicates using the TCP/IP stack. The equipment was successfully used to simulate VLSI circuits whose solution by CSM was much faster and more memory-efficient than the state-of-the-art SPICE.
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