Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Differentiable interpolants for high-order Runge-Kutta methods
SIAM Journal on Numerical Analysis
Runge-Kutta Software with Defect Control four Boundary Value ODEs
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Continuous numerical methods for ODEs with defect control
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
A BVP solver based on residual control and the Maltab PSE
ACM Transactions on Mathematical Software (TOMS)
Solving ODEs with MATLAB
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Numerical methods for both ordinary differential equations (ODEs) and delay differential equations (DDEs) are traditionally developed and assessed on the basis of how well the accuracy of the approximate solution is related to the specified error tolerance on an adaptively-chosen, discrete mesh. This may not be appropriate in numerical investigations that require visualization of an approximate solution on a continuous interval of interest (rather than at a small set of discrete points) or in investigations that require the determination of the 'average' values or the 'extreme' values of some solution components. In this paper we will identify modest changes in the standard error-control and stepsize-selection strategies that make it easier to develop, assess and use methods which effectively deliver approximations to differential equations (both ODEs and DDEs) that are more appropriate for these type of investigations. The required changes will typically increase the cost per step by up to 40%, but the improvements and advantages gained will be significant. Numerical results will be presented for these modified methods applied to two example investigations (one ODE and one DDE).