Per Grove Thomsen

Interpolants for Runge-Kutta formulas

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 interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>5</supscrpt>) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with <italic>O</italic>(<italic>h</italic><supscrpt>6</supscrpt>) local truncation error for this RK formula.

Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants

An automatic technique for solving discontinuous initial-value problems is developed and justified. The technique is based on the use of local interpolants such as those that have been developed for use with Runge-Kutta formula pairs. Numerical examples are presented to illustrate the significant improvement in efficiency and reliability that results when this technique is used with standard methods.

An ESDIRK method with sensitivity analysis capabilities

Abstract 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 information already computed for the state integration, hereby minimizing the extra effort required for sensitivity integration. Through case studies the new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present, which renders the new algorithm particularly suitable for dynamic optimization purposes.

Embeddedsdirk-methods of basic order three

The purpose of this report is to construct 3-stage SDIRK-methods (Singly Diagonally Implicit Runge-Kutta) to be used for the code SIMPLE. The local error control is performed by embedding techniques. Pairs with and without extrapolation are given.

Non-smooth Problems in Vehicle Systems Dynamics

The vehicle systems are modelled mathematically as parameter dependent multi-body systems. The connections between the elements are formulated either as dynamical equations or algebraic, or transcendental or tabulated constraint relations. The connections can rarely be modelled by analytic functions, and the missing analyticity can arise from non-uniqueness or discontinuities in the functions themselves or in their derivatives of any order. In vehicle systems the contact between the vehicle and its support (road or rail) is an important source of missing analyticity. The suspension systems of the vehicles consist of passive and active elements such as springs, dampers and actuators, and their characteristics are only analytic functions within certain intervals of operation. Unilateral contacts in the suspension systems may give rise to changes of the degrees of freedom of the system during operation, and cause impacts or sliding contact during the operation. 1 General Vehicle Model Figure 1 shows a typical 4-axle railway passenger car. The car body rests on two 2-axle carriages called bogies (bougies) or in USA trucks. The entire suspension system is built into the bogies. Fig. 1 A railway passenger car with a car body on two bogies H. True (B) DTU Informatics, The Technical University of Denmark, Kgs.Lyngby, Denmark e-mail: ht@imm.dtu.dk P.G. Thomsen, H. True (eds.), Non-smooth Problems in Vehicle Systems Dynamics, DOI 10.1007/978-3-642-01356-0 1, C

Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations

Two-parameter families of predictor-corrector methods based upon a combination of Adams- and Nyström formulae have been developed. The combinations use correctors of order one higher than that of the predictors. The methods are chosen to give optimal stability properties with respect to a requirement on the form and size of the regions of absolute stability. The optimal methods are listed and their regions of absolute stability are presented. The efficiency of the methods is compared to that of the corresponding Adams methods through numerical results from a variable order, variable stepsize program package.

Automatic Solution of Differential Equations Based on the User of Linear Multistep Methods

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 depends not so much on the accuracy requirements but rather on the absolute stabdlty properties of the formulas included in the particular VSVFM. Therefore, at least for problems where the absolute stability requirements dominate the accuracy requirements, it Is unportant to use only formulas with the best possible absolute stability characteristics in the VSVFM. Moreover, it is important to find an algorithm which predicts the largest possible stepslzes so that the next steps will be successful (the local truncation error estimator is enurely unable to do this when the absolute stability requirements are dominant). An attempt to use formulas with large absolute stability regions and to apply a strategy which normally will ensure stable computations is discussed.

Control volume based modelling in one space dimension of oscillating, compressible flow in reciprocating machines

Abstract We present an approach for modelling unsteady, primarily one-dimensional, compressible flow. The conservation laws for mass, energy, and momentum are applied to a staggered mesh of control volumes and loss mechanisms are included directly as extra terms. Heat transfer, flow friction, and multidimensional effects are calculated using empirical correlations. Transformations of the conservation equations into new variables, artificial dissipation for dissipating acoustic phenomena, and an asymmetric interpolation method for minimising numerical diffusion and non physical temperature oscillations are presented. The capabilities of the approach are illustrated with an example solution and an experimental validation of a Stirling engine model.

Adaptive Stepsize Control in Implicit Runge-Kutta Methods for Reservoir Simulation

Abstract 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 control of the integrated iterative solver is presented. We try to improve the predictive stepsize control by smoothing the stepsize sequence through combining the control of error with the control of convergence.

SENSITIVITY ANALYSIS IN INDEX-1 DIFFERENTIAL ALGEBRAIC EQUATIONS BY ESDIRK METHODS

Abstract 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 equations. The algorithm is a one-step method which makes it especially efficient compared to multiple-step methods when frequent discontinuities are present. These advantages render the new algorithm particularly suitable for dynamic optimization and nonlinear model predictive control. The algorithm is tested on the Dow Chemicals benchmark problem.