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Solving ode initial value problems with implicit taylor series methods
| Content Provider | NASA Technical Reports Server (NTRS) |
|---|---|
| Author | Scott, James R. |
| Copyright Year | 2000 |
| Description | In this paper we introduce a new class of numerical methods for integrating ODE initial value problems. Specifically, we propose an extension of the Taylor series method which significantly improves its accuracy and stability while also increasing its range of applicability. To advance the solution from t (sub n) to t (sub n+1), we expand a series about the intermediate point t (sub n+mu):=t (sub n) + mu h, where h is the stepsize and mu is an arbitrary parameter called an expansion coefficient. We show that, in general, a Taylor series of degree k has exactly k expansion coefficients which raise its order of accuracy. The accuracy is raised by one order if k is odd, and by two orders if k is even. In addition, if k is three or greater, local extrapolation can be used to raise the accuracy two additional orders. We also examine stability for the problem y'= lambda y, Re (lambda) less than 0, and identify several A-stable schemes. Numerical results are presented for both fixed and variable stepsizes. It is shown that implicit Taylor series methods provide an effective integration tool for most problems, including stiff systems and ODE's with a singular point. |
| File Size | 2613633 |
| File Format | |
| Language | English |
| Publisher Date | 2000-03-01 |
| Access Restriction | Open |
| Subject Keyword | Numerical Analysis Extrapolation Boundary Value Problems Accuracy Newton Methods Differential Equations Problem Solving Taylor Series Numerical Stability Ntrs Nasa Technical Reports Server (ntrs) Nasa Technical Reports Server Aerodynamics Aircraft Aerospace Engineering Aerospace Aeronautic Space Science |
| Content Type | Text |
| Resource Type | Technical Report |
