Mathematical modelling of problems of mathematical physics with periodic boundary conditions
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Latvijas Universitāte
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eng
Abstract
Darbā izstrādāti jauni speciāli algoritmi parasto un parciālo diferenciālvienādojumu problēmu ar periodiskajiem nosacījumiem skaitliskai modelēšanai, kuri balstās uz precīzā spektra izmantošanu telpisko parciālo atvasinājuma aproksimēšanai ar galīgajām diferencēm. Algoritmi tiek veidoti dažādām divdimensiju matemātiskās fizikas problēmām (lineārām un nelineārām), balstoties uz taišņu metodes algoritmiem un precīzā spektra diferenču shēmām. Izveidotie algoritmi tiek realizēti un salīdzināti ar datorprogrammas MATLAB palīdzību. Ar iegūtajiem algoritmiem tiek risinātas vairākas lietišķas problēmas, t.sk 2D magneto-hidrodinamiska plūsma ap periodiski novietotiem cilindriem, 2D plūsma cilindrā ārējā magnētiskā lauka ietekmē un metāla koncentrācija kūdras slāņos.
In this work new special algorithms are developed for ordinary and partial differential equation problems with periodic boundary conditions for numerical modeling. These algorithms are based on exact spectrum usage for spatial approximation of partial derivative and method of finite differences. Algorithms are shown for different types of two dimensional problems of mathematical physics, linear and nonlinear, basing on the method of lines and difference schemes with exact spectrum. Created algorithms are realized and results are compared using the program MATLAB. With the implemented algorithms several applied problems are solved, i.e. the 2D magnetohydrodynamic flow around cylinders placed periodically, 2D flow inside the cylinder depending on the external magnetic field and the metal distribution in peat layers.
In this work new special algorithms are developed for ordinary and partial differential equation problems with periodic boundary conditions for numerical modeling. These algorithms are based on exact spectrum usage for spatial approximation of partial derivative and method of finite differences. Algorithms are shown for different types of two dimensional problems of mathematical physics, linear and nonlinear, basing on the method of lines and difference schemes with exact spectrum. Created algorithms are realized and results are compared using the program MATLAB. With the implemented algorithms several applied problems are solved, i.e. the 2D magnetohydrodynamic flow around cylinders placed periodically, 2D flow inside the cylinder depending on the external magnetic field and the metal distribution in peat layers.