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The Development of System-level Solver of Dynamic Differential Equations

Author: YuPeng
Tutor: YaoWeiAn
School: Dalian University of Technology
Course: Solid Mechanics
Keywords: Runge-Kutta method Precise integration method Differential Equations Differential-algebraic equations
CLC: O175
Type: Master's thesis
Year: 2010
Downloads: 91
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Abstract


This paper is summarized and refined in ordinary differential equations and numerical solution of differential-algebraic equations , based on the use of C language program development related algorithms and implementation . Testing a large number of numerical examples show that the thesis developed by the four solvers can well meet the actual needs of scientific computing projects . The algorithm implemented in this thesis include: ( a ) non-rigid differential equations 4 and 8th-order explicit Runge-Kutta method ; ( 2 ) linear differential equations with constant coefficients precise integration method ; ( 3 ) stiff differential equations and differential algebraic equations 5 order implicit Runge-Kutta method . In order to improve efficiency and solve solver precision solver developed by the calculation carried out a detailed step size selection control, including automatic initial step size selection mechanism , adaptive variable step size selection mechanism , automatic rigid determine mechanisms and Newton iteration automatic step size adjustment mechanism. A large number of tests proved that the selected step size selection mechanism allows four solvers both in accuracy and in efficiency have been significantly improved , can be widely used system dynamics simulation analysis numerical analysis . Tests for non- stiff differential equations show that the R \u0026 D 4 and 8th-order Runge-Kutta method explicit solver can give very good numerical results . Compared two methods : 8 order explicit Runge-Kutta method to solve solver both in efficiency or accuracy in the solution to be generally higher than the fourth-order Runge-Kutta method explicit solver , which is due 8 order explicit Runge- Kutta method is a higher-order approximation , but the eight -order Runge-Kutta method explicit solver requires more memory space. Problems for stiff differential equations and differential-algebraic problems , using explicit Runge-Kutta method is unable to obtain accurate numerical results , we must use an implicit Runge-Kutta method or the precise integration method to solve . A large number of tests proved that the problem for stiff differential equations , developed by the precise integration method solver and fifth-order Runge-Kutta method implicit solver can give high precision numerical solution ; rather common for engineering differential-algebraic problems , fifth-order implicit Runge-Kutta method solver can also give high precision numerical solution . This thesis work by the National High Technology Research and Development Program (No.2009AA044501) and the Liaoning Provincial Higher research plan (No.2009S018) funding.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations
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