Global Error Control Research Articles

Benchmark results in computational heat and fluid flow: The integral transform method

The integral transform method is reviewed as a benchmark tool in computational heat and fluid flow, with special emphasis on nonlinear problems. The hybrid numerical-analytical nature of this approach collapses the numerical task into one single independent variable, and thus allows for a simple computational procedure with automatic global error control and mild increase in computational effort for multidimensional situations. Various applications on nonlinear diffusion and convection-diffusion are more closely considered, followed by sample results from recent contributions.

Global error control for the continuous Galerkin finite element method for ordinary differential equations

Une methode d'elements finis de type Galerkin continue pour l'integration des problemes initiaux pour les equations aux derivees ordinaires est analysee. Des estimations d'erreur quasi-optimales de type a priori et a posteriori sont demontrees. Les resultats sont employes dans la construction d'une theorie rigoureuse et robuste pour le controle global d'erreur. La qualite du controle d'erreur est exposee dans une serie d'experiences numeriques

Mathematical software for Sturm-Liouville problems

Software is described for the Sturm-Liouville eigenproblem. Eigenvalues, eigenfunctions, and spectral density functions can be estimated with global error control. The method of approximating the coefficients forms the mathematical basis. The underlying algorithms are briefly described, and several examples are presented.

Numerical solution of retarded initial value problems: Local and global error and stepsize control

Retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme. Hence one can control the global error of the modified problem but not directly the actual global error of the original problem. In this paper we give an estimate for the actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems. Along with the new definition we are led to developing a reliable basis for a step selection scheme.

Double Integration Using One-Dimensional Adaptive Quadrature Routines: A Software Interface Problem

A software interface problem occurs when two or more items of software are used in conjunction with one another. If proper advantage of using good software is to be gained, the user has to connect them properly. In this paper the problem of double integration employing two similar one-dimensional adaptive quadrature routines is considered. If an absolute error no greater than epsilon/sub T/ in the final result is desired, it is necessary to determine what tolerance parameter epsilon/sub o/ to assign to the outer routine and what tolerance parameters epsilon/sub Ii/ to assign for calls to the inner routine. Since it is not known at the outset how many times the inner routine will be calledm there is required what is termed an accuracy-assignment strategy for determining epsilon/sub iI/. In this paper two simple accuracy assignment strategies are discussed in detail. It is shown that one of them can be unstable if the outer routine is constructed internally by one way (local error control), but that it is quite stable if it is constructed in another way (global error control). It is also found that one of the strategies is marginally more efficient for visually smooth integrands, while the othermore » is significantly more efficient for peaked integrands.« less

New Runge-Kutta algorithms for numerical simulation in dynamical astronomy

Some new Runge-Kutta and Runge-Kutta-Nystrom algorithms are presented for the solution of ordinary differential equations of the initial value type. The methods are compared with others in integrating the equations of motion of the two body problem and are shown to offer advantages in efficiency. It is also demonstrated that the new methods can be ‘tuned’ to achieve some measure of global error control.