Module rgsl::types::ordinary_differential_equations
[−]
[src]
Numerical ODE solvers.
Ordinary Differential Equations
This chapter describes functions for solving ordinary differential equation (ODE) initial value problems. The library provides a variety of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines, and higher-level components for adaptive step-size control. The components can be combined by the user to achieve the desired solution, with full access to any intermediate steps. A driver object can be used as a high level wrapper for easy use of low level functions.
Defining the ODE System
The routines solve the general n-dimensional first-order system,
dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t)) for i = 1, \dots, n. The stepping functions rely on the vector of derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j. A system of equations is defined using the gsl_odeiv2_system datatype.
Stepping Functions
The lowest level components are the stepping functions which advance a solution from time t to t+h for a fixed step-size h and estimate the resulting local error.
Adaptive Step-size Control
The control function examines the proposed change to the solution produced by a stepping function and attempts to determine the optimal step-size for a user-specified level of error.
Evolution
The evolution function combines the results of a stepping function and control function to reliably advance the solution forward one step using an acceptable step-size.
Driver
The driver object is a high level wrapper that combines the evolution, control and stepper objects for easy use.
References and Further Reading
Ascher, U.M., Petzold, L.R., Computer Methods for Ordinary Differential and Differential-Algebraic Equations, SIAM, Philadelphia, 1998. Hairer, E., Norsett, S. P., Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, 1993. Hairer, E., Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996. Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,
Abramowitz & Stegun (eds.), Handbook of Mathematical Functions, Section 25.5. The implicit Bulirsch-Stoer algorithm bsimp is described in the following paper,
G. Bader and P. Deuflhard, “A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations.”, Numer. Math. 41, 373–398, 1983. The Adams and BDF multistep methods msadams and msbdf are based on the following articles,
G. D. Byrne and A. C. Hindmarsh, “A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations.”, ACM Trans. Math. Software, 1, 71–96, 1975. P. N. Brown, G. D. Byrne and A. C. Hindmarsh, “VODE: A Variable-coefficient ODE Solver.”, SIAM J. Sci. Stat. Comput. 10, 1038–1051, 1989. A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers.”, ACM Trans. Math. Software 31, 363–396, 2005.
Structs
| ODEiv2Control | |
| ODEiv2ControlType | |
| ODEiv2Driver | |
| ODEiv2Evolve | |
| ODEiv2Step | |
| ODEiv2StepType | |
| ODEiv2System |
Description of a system of ODEs. |