The Development Of System-level Solver Of Dynamic Differential Equations

After reviewing on the numerical solution of ordinary differential equations and differential algebraic equations, some solvers of ordinary differential equations and differential algebraic equations are implemented with C language. A large number of tests show that the solvers to describe in this thesis are able to satisfy well the practical demand of engineering and scientific computing.The algorithms to describe in this thesis includes:(1) 4-order and 8-order explicit Runge-Kutta method for non-stiff ordinary differential equations; (2) precise integration method (PIM) for linear differential equations; (3) 5-order implicit Runge-Kutta method for stiff differential equations and differential algebraic equations. To improve efficiency and precision of the solvers, selection of step size in the solvers is controlled choicely, which includes automatic selection mechanism for the initial step, the mechanism of adaptive variable step size selection, the mechanism of automatic estimation for stiff equations and automatic adjusting mechanism for the step size of Newton iteration. A large amount of tests show that the solvers with these selected mechanisms on step size have been improved observably both in accuracy and in efficiency, and can be widely applied in the numerical analysis of dynamic simulation systems.The tests on non-stiff differential equations show that, both 4-order and 8-order explicit Runge-Kutta method can give very accurate numerical results. Relatively, the solver of 8-order explicit Runge-Kutta method is generally higher than the solver of 4-order explicit Runge-Kutta method both in accuracy and efficiency due to approximation order of 8-order explicit Runge-Kutta method is higher. However, more memory space is needed when 8-order explicit Runge-Kutta method is used.For stiff differential equations and differential-algebraic equations, the explicit Runge-Kutta method is not able to give precise numerical results, so implicit Runge-Kutta method or PIM must be used. A greal deal of tests prove that, both the solver of PIM and the solver of 5-order implicit Runge-Kutta method can give the high precision numerical solution of stiff differential equations, and the solver of 5-order implicit Runge-Kutta method can also give the high precision numerical solution of common differential-algebraic equations.The work described in this thesis was supported by the National High Technology Research and Development Program of China (No.2009AA044501) and Higher Scientific Research Program of Liaoning Province of China (No.2009S018).