Basem S. Attili

An efficient implicit Runge–Kutta method for second order systems

Abstract We will consider the efficient implementation of a fourth order two stage implicit Runge–Kutta method to solve periodic second order initial value problems. To solve the resulting systems, we will use the factorization of the discretized operator. Such proposed factorization involves both complex and real arithmetic. The latter case is considered here. The resulting system will be efficient and small in size. It is one fourth the size of systems using normal implicit Runge–Kutta method. Numerical details and examples will also be presented to demonstrate the efficiency of the method.

Superconvergence and the computation of generalized turning points for B.V.P’s

Abstract We consider projection methods in general and collocation at gauss points in particular for the numerical computation of generalized turning points for two point B.V.P’s. We discretize the original problem using collocation to obtain a finite dimensional problem. We will employ a direct method for the characterization and computation of simple turning points. Proper extensions of this direct method will be used for the computation of bifurcation points and cubic turning points. We comment also on the superconvergence which will be clear from the numerical examples.

Initial value methods for the primary two point boundary value problems in modeling viscoelastic flows

Initial value solvers through shooting methods will be used to compute solutions to the primary two-point boundary value problem arising in modeling viscoelastic flow. We will show that the classical Runge-Kutta methods will have at least h 2-order of convergence due to the presence of the singularity. Comparison with the work of others will also be presented through some numerical results and examples.

On the use of predictor-corrector continuation to trace implicitly defined curves and calculating bifurcation

We will consider the use of Predictor-corrector method to trace parameterized curves. Homotopy methods will be needed since a Newton like method cannot be used to solve the nonlinear systems involved. This is due to the fact that not much information is available about the zero point of the system. We will also consider systems which involve the presence of a natural parameter; in particular, tracing a parameter dependent curve which contains-a simple turning or bifurcation point at a critical value of the parameter.

Numerical continuation through folds with step control

We consider an efficient algorithm for tracing implicitly defined curves. The algorithm does not parametrize the solution in terms of arc length or in terms of the naturally occurring parameter λ, but rather does a parameter transformation that chooses the component relatively contributing the most change as the parameter. For comparison purposes, we present the predictor–corrector path following technique. The exact computation of fold points of the simple turning point type will also be presented. Numerical examples that demonstrate the efficiency of the algorithm will be done.

On the numerical implementation of the shooting methods to the one-dimensional singular boundary value problems

Simple and multiple shooting methods are proposed to numerically solving singular boundary value problems with a regular singularity at one end of the interval. The singularity is first removed using series solution in the vicinity of the singular point to produce a regular boundary value problem. Numerical examples with some comparison of the work of others are also included.

Numerical Treatment of Differential‐Algebraic Equations with Index 2

We will consider index‐2 differential algebraic systems. Since they are usually harder to solve, we will show how to reduce the index 2 problem to index 1 DAE which becomes easier to solve numerically. For the numerical treatment, we will treat the resulting index‐1 DAE using power series solutions coupled with pade approximation for better convergence results. Numerical examples will be presented also.