David C. Williams
ABSTRACT
Many routines for solving differential equations require the user to specify an estimate of the initial step size. We introduce a means of estimating an appropriate value automatically. Strong emphasis is placed on the need for scale invariance in making such estimates, and we therefore review the importance of scale and matters equivalent to scale in estimating the order of magnitude of the appropriate starting step size. In this context, we also discuss the response to scale of integrators employing the error per unit step concept of error control and are forced to conclude that our procedure should not be used in such integrators, if the latter should be used at all. We note a special case where the procedure could be badly deceived and suggest an option by which the user can protect himself against this kind of mishap. Test results are cited.
- Fred T. Krogh, "On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations," J. ACM 20 (1973), 545--562. Google Scholar
Digital Library
- L. F. Shampine, "Limiting Precision in Differential Equation Solvers," Math. Comput. 28 (1974), 141--144.Google ScholarCross Ref
- R. W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill, New York, 1962. Google ScholarDigital Library
- D. C. Williams, "The Application of Predictor-Corrector Algorithms to the Direct Integration of Second Order Ordinary Differential Equations," SLA-73-0058, June 1974.Google Scholar
Recommendations
Implicit-Explicit Strong Stability Preserving Runge-Kuta Methods with High Linear Order
PEARC '17: Proceedings of the Practice and Experience in Advanced Research Computing 2017 on Sustainability, Success and ImpactHigh order strong stability preserving (SSP) time discretizations have been extensively used with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. Explicit SSP Runge--Kutta methods exist only up to fourth ...
High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws
Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large ...
Implicit and Implicit---Explicit Strong Stability Preserving Runge---Kutta Methods with High Linear Order
Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. The search for high order strong ...
Comments