Selection Strategies for Set-Valued Runge-Kutta Methods
نویسنده
چکیده
A general framework for proving an order of convergence for set-valued Runge Kutta methods is given in the case of linear differential inclusions, if the attainable set at a given time should be approximated. The set-valued method is interpreted as a (set-valued) quadrature method with disturbed values for the fundamental solution at the nodes of the quadrature method. If the precision of the quadrature method and the order of the disturbances fit together, then an overall order of convergence could be guaranteed. The results are applied to modified Euler method to emphasize the dependence on a suitable selection strategy (one strategy leads to an order breakdown).
منابع مشابه
Analysis of Stepsize Selection Schemes for Runge - Kutta Codes
Conditions on Runge-Kutta algorithms can be obtained which ensure smooth stepsize selection when stability of the algorithm is restricting the stepsize. Some recently derived results are shown to hold for a more general test problem.
متن کاملApplicability of Load Balancing Strategies to Data-Parallel Embedded Runge-Kutta Integrators
Embedded Runge-Kutta methods are among the most popular methods for the solution of non-stiff initial value problems of ordinary differential equations (ODEs). We investigate the use of load balancing strategies in a dataparallel implementation of embedded Runge-Kutta integrators. Since the parallelism contained in the function evaluation of the ODE system is typically very fine-grained, our ai...
متن کاملAn overview of high-order finite difference schemes for computational aeroacoustics
One of the problems in computational aeroacoustics (CAA) is the large disparity between the length and time scales of the flow field, which may be the source of aerodynamically generated noise, and the ones of the resulting acoustic field. This is the main reason why numerical schemes, used to calculate the timeand space-derivatives, should exhibit a low dispersion and dissipation error. This p...
متن کاملA new higher-order weak approximation scheme for stochastic differential equations and the Runge-Kutta method
The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge–Kutta method for ODEs is directly applicable...
متن کاملA Lyapunov exponents based stability theory for ODE initial value problem solvers
In this paper we consider the stability of variable step-size Runge-Kutta methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees the numerical solution of an asymptotically decaying time-dependent linear problem also de...
متن کامل