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A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an… (More)

Variable-stepsize variable-formula methods (VSVFMs) are often used in the numerical integration of systems of ordinary differential equations. In this way, roughly speaking, one attempts to minimize the number of steps, Le., to select the largest possible stepsize according to a prescribed error tolerance. Very often, however, the selectmn of the stepsize… (More)

A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge–Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. A key feature is the reuse of… (More)

- Carsten Völcker, John Bagterp Jørgensen, Per Grove Thomsen, Erling Halfdan Stenby
- 2010

This paper concerns predictive stepsize control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are described. A predictive stepsize adjustment rule based on error estimates and convergence… (More)

- Morten Rode Kristensen, John Bagterp Jørgensen, Per Grove Thomsen, Michael Locht Michelsen, Sten Bay Jørgensen
- 2005

Dynamic optimization by multiple shooting requires integration and sensitivity calculation. A new semi-implicit Runge-Kutta algorithm for numerical sensitivity calculation of index-1 DAE systems is presented. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity… (More)

It is important to incorporate all available observations when large-scale mathematical models arising in different fields of science and engineering are used to study various physical and chemical processes. Variational data assimilation techniques can be used in the attempts to utilize efficiently observations in a large-scale model (for example, in order… (More)