We continue to study the functioning of the human lung. Earlier, an automaton model has been constructed of self-cleaning of dust-loaded lungs in a pure environment. In this paper, we study the Moore diagram of the suitable automaton. As the characteristic states of the diagram we choose the states with no predecessors, such states are called start ones. We study the start states of the Moore diagram.
We construct an automaton model of lung self-cleaning mechanism both in the pure environment and in the conditions of possible dust loading into the lungs from the environment in breathing. We analyse this model and find the complexity (time) of self-cleaning of the lungs for an arbitrary initial dust load.
It is well known that the theory of asymptotic expansion of the global error of one-step methods is an important but complicated fact in the realm of numerical analysis of differential equations. It can hardly be confirmed for majority of numerical schemes and problems by practical computations. On the other hand, this theory is a fundamental tool to justify the extrapolation technique, which is one of the most efficient means to solve ordinary differential equations. Therefore, in the recent paper Kulikov [Numer. Algorithms 53: 321–342, 2010] presented his local theory of extrapolation methods, which is based on the Richardson technique only and does not require any asymptotic global error expansion. Here, we concentrate on quadratic extrapolation. We explain which property of symmetric one-step methods provides two-order growth of accuracy of the underlying method after each extrapolation step and arrive at the notion of proportional extrapolation. We also learn more about adjoint and symmetric one-step methods. In addition, we prove that the modified Aitken–Neville algorithm works for any symmetric one-step method of an arbitrary order 2s.
Five different error estimation strategies suitable for the Gauss-type Nested Implicit Runge–Kutta method of order 4 have been presented and tested numerically in [Kulikov and Shindin, Lecture Notes in Computer Science: 136–143, 2007, Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. The nested Implicit Runge–Kutta schemes introduced recently are an efficient class of Implicit Runge–Kutta formulas. In this paper we deal with the methods of order 6. One scheme of such sort has been constructed in [Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. Now we present a one-parametric family of the above-mentioned formulas of order 6 by relaxing the accuracy requirement for some stage values. This allows the error estimation strategies designed for the method of order 4 to be extended to the higher-order Gauss-type Nested Implicit Runge–Kutta method. We also present the particulars of the efficient implementation of this method, which is stable and accurate. The numerical examples confirm the efficiency of the numerical scheme under consideration for both ordinary differential equations and partial differential equations.
We develop the base principles of the theory of implicit extrapolation methods for ordinary differential equations. Of considerable importance for the practical implementation of these methods are the estimates of a sufficient number of iterations in an iteration method used to solve the system of algebraic equations, which results from the discretization of the original problem. In this paper we consider three best known iterative processes, viz. the simple iteration method, the generalized and modified Newton methods for which we prove the estimates of a sufficient number of iterations when extrapolating the numerical solution.
Naturally, implicit extrapolation methods are very time-consuming. However, if they are based on the implicit Runge-Kutta methods with appropriate stability properties, the algorithms thus obtained are applicable for solving a broader class of problems. Moreover, in this paper we construct the theory of minimum implicit one-step methods whose extrapolation process requires the least (in the class of implicit methods) computer time. We prove that the above estimates of a sufficient number of iterations allow us to correctly realize implicit quadratic extrapolation in practice. The efficiency of the new class of methods and the importance of the sufficient estimates of the number of iterations for implicit extrapolation methods are demonstrated with numerical examples.
In this paper we first formulate and prove a number of theorems concerning the convergence
of combined numerical one-step methods for index 1 differential-algebraic systems. Then,
we use these results to justify an implicit extrapolation technique and show their practical importance.
Second, we give a theory of adjoint and symmetric one-step methods for differential-algebraic
equations and we also determine symmetric methods among Runge–Kutta formulae. We prove that algebraically
stable symmetric Runge–Kutta formulae are symplectic and they have a structure which is
in some sense similar to the structure of Gauss methods. Finally, we come to the concept of quadratic
extrapolation for index 1 differential-algebraic systems and develop an advanced version of the localglobal
step size control based on the extrapolation technique. Numerical tests support the theoretical
results of the paper.
In this paper we find and study the class of symmetric methods among the Runge-Kutta
formulae. It is shown that the explicit Runge-Kutta methods cannot be symmetric. We also define the
conditions which coefficients in the implicit Runge-Kutta method should satisfy for it to be symmetric.
Particular attention has been given to the study of stability properties in the symmetric Runge-Kutta
formulae. It is proved that in some cases the notions of absolute and algebraic stability for the given
class of numerical methods coincide. Besides, we find a restriction to the order of stable symmetric
methods among the diagonally implicit Runge-Kutta formulae. Finally, we give full characteristics of
all algebraically stable symmetric Runge-Kutta methods in terms of a transformed matrix of coefficients.