Large Regions Of Absolute Stability Research Articles

Explicit Nordsieck methods with extended stability regions

We describe the construction of explicit Nordsieck methods of order p and stage order q = p with large regions of absolute stability. We also discuss error propagation and estimation of local discretization errors. The error estimators are derived for examples of general linear methods constructed in this paper. Some numerical experiments are presented which illustrate the effectiveness of proposed methods.

Explicit Nordsieck methods with quadratic stability

We describe the construction of explicit Nordsieck methods with s stages of order p?=?s???1 and stage order q?=?p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge---Kutta stability.

On a class of spline-collocation methods for solving second-order initial-value problems

We propose global collocation methods for second-order initial-value problems y″=f(x, y) and y″=f(x, y, y′). The present methods are based on quintic C2-splines S(x) with three collocation points , j=1, …, 3 in each subinterval [x i−1, x i ], i=1, …, N. It is shown that the method (c 1=(5−√5)/10, c 2=(5+√ 5)/10) has a convergence of order six, while in the remaining cases (c 1, c 2∈(0, 1), with c 1≠c 2) the order is five. The absolute stability properties appear that for all c 1, c 2∈[0.8028, 1) with c 1≠c 2, the methods are A-stable independent of the particular choice of the collocation points, while the sixth-order method has a large region of absolute stability. Moreover, the sixth-order method has a phase-lag of order six with actual phase-lag (3/25(8!))v 6, and it possesses (0, 37.5)∪(60, 122.178) as the interval of periodicity and absolute stability. The superiority of the obtained methods is demonstrated by considering periodic stiff problems of practical interest.

Construction of two-step block Simpson type method with large region of absolute stability

In this note, we describe the construction of two-step Block Simpson type method with non-uniform order, in continuous approximation form, with large regions of absolute stability. All the discrete schemes used in the block method come from a single continuous formulation. When used in block form, the suggested approach is self- starting, accurate and efficient. Numerical result is included to further justify our present method. JONAMP Vol. 11 2007: pp. 261-268

Construction of two-step Runge–Kutta methods with large regions of absolute stability

We describe the construction of explicit two-step Runge–Kutta methods of order p and stage order q= p−1 or q= p with large regions of absolute stability. This process is illustrated for the methods of order p=2, and 3 and leads to new methods which are competitive with explicit Runge–Kutta methods of the same order.

A Fifth-Order Exponentially Fitted Formula

A fifth-order second derivative formula is developed for stiff systems of ordinary differential equations. The exponentially fitted formulas possess a large region of absolute stability, and numerical results demonstrate increased accuracy with the same computational effort when compared with similar fourth-order formulas.

High order methods for the numerical integration of ordinary differential equations

High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.