A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations

@article{Byrne1975APF,
  title={A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations},
  author={George D. Byrne and Alan C. Hindmarsh},
  journal={ACM Trans. Math. Softw.},
  year={1975},
  volume={1},
  pages={71-96}
}
Two variable-order, varlable-step size methods for the numerical solution of the initial value problem for ordinary differential equations are presented. These methods share a common philosophy and have been combined in a single program. The two integrators are for stiff and nonstiff ordinary differential equations, respectively. The former integrator is based on backward differentiation formulas of orders one through five, each of which is stiffly stable, while the latter is based on formulas… 
View on ACM
doi.org
266 Citations
Highly Influential Citations
14
Background Citations
30
Methods Citations
59
Results Citations
1

Figures and Tables from this paper

266 Citations

A composite integration scheme for the numerical solution of systems of ordinary differential equations
An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs
TLDR
An alternative technique for the implementation of variable step-size multistep formulas is developed for the numerical solution of ordinary differential equations and results indicate that methods based on this technique have stability properties similar to those of the corresponding variable coefficient implementations.
Comparing Numerical Methods for the Solution of Stiff Systems of ODEs Arising in Chemistry
Comparison of Numerical Methods for Solving Initial Value Problems for Stiff Differential Equations
TLDR
This study has focused on some conventional methods namely Runge-Kutta method, Adaptive Stepsize Control for Runge’s Kutta and an ODE Solver package, EPISODE and describes the characteristics shared by these methods.
Initial Value Problems
This chapter is devoted to initial values problems for ordinary differential equations. It discusses theory for existence, uniqueness and continuous dependence on the data of the problem. Special…
Improving the Efficiency of Matrix Operations in the Numerical Solution of Stiff Ordinary Differential Equations
TLDR
This updating technique is shown to be applicable to a wide variety of methods for stiff systems including multistep methods, Runge-Kutta methods, and methods using a rational function of a matrix.
Blended Linear Multistep Methods
TLDR
A variable-order, variable-step blend of the Adams-Moulton and the backward differentiation formulas, similar to Lambert and Sigurdsson's linear multistep formulas with variable matrix coefficients, but the approach is different.
Numerical Approach for Solving Stiff Differential Equations: A Comparative Study
TLDR
The aim is to identify the problem area and the characteristics of the stiff differential equations for which the equations are distinguishable, and to find that though the method is very efficient it has certain problem area for a new user.
Linearly Implicit Multistep Methods for Time Integration
TLDR
A new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep, which shows measurable performance improvement over a similar BDF implementation.
Using Diagonally Implicit Multistage Integration Methods for Solving Ordinary Differential Equations. Part 2: Implicit Methods.
TLDR
Results of successful tests on the Prothero-Robinson problem are reported and demonstrate that DIMSIMs may be used to develop efficient stiff solvers, and some new A-stable and L-stable methods are derived.
...
...

References

SHOWING 1-10 OF 72 REFERENCES
The numerical integration of ordinary differential equations
TLDR
A reliable efficient general-purpose method for automatic digital computer integration of systems of ordinary differential equations is described that incorporates automatic starting and automatic choice and revision of elementary interval size and approximately minimizes the amount of computation for a specified accuracy of solution.
Comparing Numerical Methods for Ordinary Differential Equations
TLDR
According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer, however, when function evaluated methods are relatively expensive, variable-order methods based on Adams formulas are best.
On Numerical Integration of Ordinary Differential Equations
TLDR
A reliable efficient general-purpose method for automatic digital integration of systems of ordinary differential equations that is thoroughly stable under all circumstances, approximately minimizes the amount of computation for a specified accuracy of solution, and applies to any system of differential equations with deriva- tives continuous or piecewise continuous with finite jumps.
EFFICIENT INTEGRATION METHODS FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
Linear one step methods of a novel design are given for the numerical solution of stiff systems of ordinary differential equations. These methods permit fast integration with increments h of the…
Stability and convergence of variable order multistep methods
A method based on variable step Adams formulas is shown to be stable for any order changing scheme. A method based on the Nordsieck form of Adams formulas, however, is shown to be stable only if the…
A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas
The backward differentiation formulas (BDF), of order 1 up to 6 are described as they are applied to a system of differential algebraic equations. The BDF method is compared to the Gear-Nordsieck…
A comparison of Crank-Nicolson and Chebyshev rational methods for numerically solving linear parabolic equations
The automatic integration of ordinary differential equations
An integration technique for the automatic solution of an initial value problem for a set of ordinary differential equations is described. A criterion for the selection of the order of approximation…
A comparative study of computer programs for integrating differential equations
  • P. Fox
  • Computer Science
    CACM
  • 1972
TLDR
A study comparing the performance of several computer programs for integrating systems of ordinary differential equations found that a program based on rational extrapolation showed the best performance.
Changing stepsize in the integration of differential equations using modified divided differences
TLDR
New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize forMultistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas.
...
...