| Title:
|
On those ordinary differential equations that are solved exactly by the improved Euler method (English) |
| Author:
|
Rivertz, Hans Jakob |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
29-34 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
As a numerical method for solving ordinary differential equations $y^{\prime }=f(x,y)$, the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for $f(x,y)$ to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method. (English) |
| Keyword:
|
extended Euler |
| Keyword:
|
numerics |
| Keyword:
|
ordinary differential equations |
| MSC:
|
34A99 |
| idZBL:
|
Zbl 06321145 |
| idMR:
|
MR3073013 |
| DOI:
|
10.5817/AM2013-1-29 |
| . |
| Date available:
|
2013-05-28T13:27:43Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143297 |
| . |
| Reference:
|
[1] Agarwal, R. P.: Difference equations and inequalities: Theory, methods, and applications.Pure and Applied Mathematics, 228, Marcel Dekker, 2000. Zbl 0952.39001, MR 1740241 |
| Reference:
|
[2] Butcher, J. C., Wanner, G.: Runge–Kutta methods: some historical notes.Appl. Numer. Math. 22 (1996), 113–151. Zbl 0867.65038, MR 1424293, 10.1016/S0168-9274(96)00048-7 |
| Reference:
|
[3] Carr, J. W., III, : Error bounds for the Runge–Kutta single–step integration process.J. Assoc. Comput. Mach. 5 (1958), 39–44. MR 0094908, 10.1145/320911.320916 |
| Reference:
|
[4] Cieśliński, J. L.: On the exact discretization of the classical harmonic oscillator equation.J. Differ. Equations Appl. 17 (11) (2011), 1673–1694. Zbl 1232.65175, MR 2846507, 10.1080/10236191003730563 |
| Reference:
|
[5] Cieśliński, J. L., Ratkiewicz, B.: Energy–preserving numerical schemes of high accuracy for one–dimensional Hamiltonian systems.J. Phys. A 44 (2011), 1751–8113. Zbl 1218.65144, MR 2783644, 10.1088/1751-8113/44/15/155206 |
| Reference:
|
[6] Euler, L.: Institutionum calculi integralis.Impensis Academiae Imperialis Scientiarum, 1768–1770. |
| Reference:
|
[7] Galler, B. A., Rozenberg, D. P.: A generalization of a theorem of Carr on error bounds for Runge-Kutta procedures.J. Assoc. Comput. Mach. 7 (1960), 57–60. Zbl 0096.10101, MR 0145673, 10.1145/321008.321015 |
| Reference:
|
[8] González–Pinto, S., Hernández–Abreu, D.: Global error estimates for a uniparametric family of stiffly accurate Runge–Kutta collocation methods on singularly perturbed problems.BIT 51 (1) (2011), 155–175. Zbl 1217.65158, MR 2784657, 10.1007/s10543-010-0304-2 |
| Reference:
|
[9] Hairer, E., Lubich, Ch., Roche, M.: Error of Runge–Kutta methods for stiff problems studied via differential algebraic equations.BIT 28 (1988), 678–700. Zbl 0657.65093, MR 0963310, 10.1007/BF01941143 |
| Reference:
|
[10] Ixaru, L. G., Vanden Berghe, G.: Exponential fitting.Kluwer Academic Publishers, 2004. Zbl 1105.65082, MR 2174666 |
| Reference:
|
[11] Lotkin, M.: On the accuracy of Runge–Kutta’s method.Math. Tables Aids Comput. 5 (1951), 128–133. Zbl 0044.33104, MR 0043566, 10.2307/2002436 |
| Reference:
|
[12] Mickens, R. E.: Nonstandard finite difference models of differential equations.World Scientific Publishing Co. Inc., 1994. Zbl 0810.65083, MR 1275372 |
| Reference:
|
[13] Potts, R. B.: Differential and difference equations.Amer. Math. Monthly 89 (1982), 402–407. Zbl 0498.34049, MR 0660921, 10.2307/2321656 |
| Reference:
|
[14] Ralston, A.: Runge–Kutta methods with minimum error bounds.Math. Comput. 16 (1962), 431–437. Zbl 0105.31903, MR 0150954, 10.1090/S0025-5718-1962-0150954-0 |
| Reference:
|
[15] Runge, C.: Über die numerische Auflösung von Differentialgleichungen.Math. Ann. 46 (1895), 167–178. MR 1510879, 10.1007/BF01446807 |
| Reference:
|
[16] Runge, C.: Über die numerische Auflösung von totaler Differentialgleichungen.Gött. Nachr. (1905). |
| . |
