Andrzej Marciniak

Implicit Interval Methods for Solving the Initial Value Problem

Implicit interval methods of Runge–Kutta and Adams–Moulton type for solving the initial value problem are proposed. It can be proved that the exact solution of the problem belongs to interval-solutions obtained by the considered methods. Furthermore, it is possible to estimate the widths of interval-solutions.

A survey of interval Runge-Kutta and multistep methods for solving the initial value problem

The paper is dealt with a number of one- and multistep interval methods developed by our team during the last decade. We present implicit interval methods of Runge-Kutta type, interval versions of symplectic Runge-Kutta methods and interval multistep methods of Adams-Bashforth, Adams-Moulton, Nystrm and Milne-Simpson types.

An interval version of the crank-nicolson method --- the first approach

To study the heat or diffusion equation, the Crank-Nicolson method is often used. This method is unconditionally stable and has the order of convergence O(k2+h2), where k and h are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including not only the method error, but also representation and rounding errors. Therefore, we propose an interval version of the Crank-Nicolson method from which we would like to obtain solutions including the discretization error. Applying such a method in interval floating-point arithmetic allows one to obtain solutions including all possible numerical errors. Unfortunately, for the proposed interval version of Crank-Nicolson method, we are not able to prove that the exact solution belongs to the interval solutions obtained. Thus, the presented method should be modified in the nearest future to fulfil this necessary condition. A numerical example is presented. Although in this example the exact solution belongs to the interval solutions obtained, but the so-called wrapping effect significantly increases the widths of these intervals.

A central-backward difference interval method for solving the wave equation

The paper is devoted to an interval difference method for solving one dimensional wave equation with the initial-boundary value problem. The method is an adaptation of the well-known central and backward difference methods with respect to discretization errors of the methods. The approximation of an initial condition is derived on the basis of expansion of a third-degree Taylor polynomial. The initial condition is also written in the interval form with respect to a discretization error. Therefore, the presented interval method includes all approximation errors (of the wave equation and the initial condition). The floating-point interval arithmetic is used. It allows to obtain interval solutions which contain all calculations errors. Moreover, it is indicated that an exact solution belongs to the interval solution obtained.

Interval versions of central-difference method for solving the poisson equation in proper and directed interval arithmetic

Abstract To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

Interval versions of Milne’s multistep methods

The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

Interval Difference Methods for Solving the Poisson Equation

In the paper we resemble interval difference method of second order designed by us earlier and present new, fourth order interval difference methods for solving the Poisson equation with Dirichlet boundary conditions. Interval solutions obtained contain all possible numerical errors. Numerical solutions presented confirm the fact that the exact solutions are within the resulting intervals.