| Summary: | 博士 === 國立成功大學 === 機械工程學系 === 90 === Abstract
A new numerical method, modified group preserving (MGP) scheme, can be applied to stiff initial-value, boundary value, structural dynamics and heat transfer problems for engineering applications. Generally speaking, MGP scheme stems mainly from GP scheme. The concept of group preserving (GP) scheme preserves the cone structure in the Minkowski space, has been developed by employing the Cayley map to form a Lie group transformation, and providing an explicit one-step time integrator for the general dynamical systems. This scheme has been proposed only for the solution on the initial value problems of linear or nonlinear problems. However, a modified group preserving scheme (MGPS) is proposed by considering the translation of the state variables to resolve the difficulties of small time step problem in solving the initial value problems with stiffness, the norm of state variables is near to the zero point or the norm of forcing function is very large. In this study, Burgers’ equation and its application are proposed.
Since the purpose of traditional numerical analysis is to represent the solution to actual problems, it is important that qualitative properties of the numerical solution should be consisting with consistency, stability and convergence in order to resemble those of the true solutions. Thus the present of qualitative properties of MGP scheme is also satisfy those criteria. In practical applications of MGP scheme are valid to show that three constraints, adaptive factor, half step size and cone constraint, are satisfied. Indicating continuously only monitor the adaptive factor takes about 1.0 for a very small value of fixed step size. It is yield a time forward explicit scheme for the next step solution in each time step. Therefore, each solution is not reliable to have information on the accuracy of solutions in traditional explicit numerical methods when the exact solution cannot be obtained. In the present study, employing two increment quantities to estimate the error of numerical solutions for ordinary differential equations, it shown the same order results of errors and step sizes. The important feature in the above analysis, the value of adaptive factor is large jump or small jump, indicating the location of large change for the amplitude and phase angle of response, respectively. This can further prove that the present scheme is effective in calculating strongly stiff initial value problems with stiffness ratio 10E+6.
The MGP scheme is extended to solve the boundary value problems by a shooting method. It has not a systematic way to assume the missing initial values. In the present study, a systematic way of finding assumed of the missing value of initial condition is proposed (see appendix G). Combining use of numerical method of lines and MGP scheme to heat transfer problems and Burgers’ equation with Reynolds number of 10000. Results consistently agree with the numerical solution is quite satisfactory. Theoretically, the MGP scheme in more universal and practical than that of traditional numerical methods for solving differential equations. Several numerical results is presented, it shows that the MGP scheme works very well, and the merits of the computational efficiency and accuracy of the method can be confirmed.
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