Linear Ordering Methods Research Articles

Zróżnicowanie powiatów województwa małopolskiego pod względem poziomu infrastruktury technicznej i społecznej

The paper presents results comparison of districts within Małopolskie region with respect to the chosen features of technical and social infrastructure in 2010. On the basis of the method of linear ordering of multidimensional objects the ranking arrangement has been carried out and then the division of districts into three groups: of high, moderate and low level of the infrastructure being investigated. The districts were grouped according to the level of equipment in certain items of infrastructure. For the grouping the method of vector elimination has been applied.

Sustainability of the Construction Sector in Poland in Comparison to other Economic Sectors

In various sectors in Poland, aims of sustainable development are being implemented more or less successfully. It is necessary to apply the principles of sustainable development to the human behaviour and activity of business entities, because there are more and more activities particularly harmful to the natural environment. The implemented modern technologies aimed at the elimination of (or eliminating) the negative impact of human and business activities on the natural environment, are bringing the desired results as far as the protection of individual natural resources is concerned. Inability to define the economic value of the natural resources results in their constant, excessive exploitation and degradation. However, the growing awareness of the environmental issue in Poland is resulting in the application of technologies that protect individual natural resources. The paper attempts to determine the measures of sustainable development of construction sector in the scope of economic and environmental order, and to classify all of the economic sectors in Poland in accordance with the achieved level sustainability. For this purpose one has used the numerical taxonomy methods, and in particular the methods of linear ordering of objects (that involve projection of the objects from a multidimensional space of features onto a straight line), called the methods of multidimensional comparative analysis. The measures were designed based on available statistical data from 2012. The purpose of this article is comparison the construction sector to each other and to find out which sectors are similar to each other in terms of sustainability level, and to find sectors that have managed to achieve the best results in this regard.

Strong-stability-preserving, Hermite–Birkhoff time-discretization based on [formula omitted] step methods and 8-stage explicit Runge–Kutta methods of order 5 and 4

Ruuth and Spiteri have shown, in 2002, that fifth-order strong-stability-preserving (SSP) explicit Runge–Kutta (RK) methods with nonnegative coefficients do not exist. One of the purposes of the present paper is to show that the Ruuth–Spiteri barrier can be broken by adding backsteps to RK methods. New optimal, 8-stage, explicit, SSP, Hermite–Birkhoff (HB) time discretizations of order p, p=5,6,…,12, with nonnegative coefficients are constructed by combining linear k-step methods of order (p−4) with an 8-stage explicit RK method of order 5 (RK(8, 5)). These new SSP HB methods preserve the monotonicity property of the solution and prevent error growth; therefore, they are suitable for solving hyperbolic partial differential equations (PDEs) by the method of lines. Moreover, these new HB methods have larger effective SSP coefficients and larger maximum effective CFL numbers than Huang’s hybrid methods and RK methods of the same order when applied to the inviscid Burgers equation. Generally, HB methods combined with RK(8, 5) have maximum stepsize 24% larger than HB combined with RK(8, 4).

A Continuous Formulation of A(α)-Stable Second Derivative Linear Multistep Methods for Stiff IVPs in ODEs

This paper considers family of A(α)-stable second derivative linear multistep methods of order p = k + 3 for step number k ≥ 15 for the solution of stiff IVPs in ODEs. The methods are demonstrated to be A(α)-stable for k ≥ 13. At k = 14, the method is stable but not A(α)-stable. The instability of the new methods sets in when k > 15. Numerical examples are given to demonstrate the application of the methods.

Porządkowanie liniowe obiektów z wykorzystaniem rozmytych metod AHP i TOPSIS

The aim of this paper was to investigate the applicability of the fuzzy multi-criteria linear ordering method to the construction of synthetic characteristics. Proposed approach base on two fuzzy methods: analytical hierarchy process (to calculate weight of characteristics and eliminated of unimportance characteristics) and method TOPSIS (to ranking of objects). The proposed procedure was employed to assess the socio-economic development of rural Wielkopolska seen as a collection of counties.

Strong-Stability-Preserving, K-Step, 5- to 10-Stage, Hermite-Birkhoff Time-Discretizations of Order 12

We construct optimal k-step, 5- to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5- to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spatial discretization. It is seen that the 8-step 7-stage HB methods have largest effective SSP coefficient among the HB methods of order 12 on hand. On Burgers’ equations, some of the new HB methods have larger maximum effective CFL numbers than Huang’s 7-step hybrid method of order 7, thus allowing larger step size.

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q = p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.

Global error control in implicit parallel peer methods

Recently, Schmitt, Weiner, and Erdmann have proposed an efficient family of numerical methods termed Implicit Parallel Peer (IPP) methods. They are a subclass of s -stage general linear methods of order s – 1. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff differential equations. The special construction of IPP methods allows for a parallel implementation, which is advantageous in modern high-performance computation environment. In this paper we add one more useful functionality to IPP methods, i.e. automatic global error control. We show that the global error estimation developed by Kulikov and Shindin in multistep formulas is suitable for the methods of Schmitt, Weiner and Erdmann. Moreover, that global error estimation can be done in parallel. An algorithm of efficient stepsize selection is also discussed here.

Order conditions for integrators and mappings with applications

We regard the elementary weight function as a mapping from input to output, so that the mappings allow us to derive the order conditions for a given method more easily and concisely. This paper presents the order conditions of one-step Runge–Kutta methods, Runge–Kutta-type methods with derivatives, a class of two-step Runge–Kutta methods and three-step Runge–Kutta methods, based on the elementary weight mappings, the composition formula of two mappings as well as the mappings I , D r , E ( ν ) defined on the set T of all rooted trees from an input to an output. It is pointed out that the order conditions can be easily achieved for linear multistep methods, and second-derivative linear multistep methods of order p , and it suffices to consider only bushy trees ▪ of order up to p together with the empty tree 0̸ for a linear multistep method, or a second-derivative linear multistep method of order p .

Construction of General Linear Methods with Runge–Kutta Stability Properties

We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one real parameter which is related to the error constant of the method. Examples of methods are derived which illustrate the application of the approach presented in this paper.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81–101] were able to construct strong local order 1.5 stochastic Runge–Kutta methods for certain cases, it is known that all extant stochastic Runge–Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler–Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649–673] (applied to deterministic non-commutative linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed – hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number ( d) of Wiener processes it is shown that strong local order 1 Runge–Kutta methods with d+1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches.

A nonlinear optimization approach to the construction of general linear methods of high order

We describe the construction of diagonally implicit multistage integration methods of order and stage order p = q = 7 and p = q = 8 for ordinary differential equations. These methods were obtained using state-of-the-art optimization methods, particularly variable-model trust-region least-squares algorithms.

Stability of some boundary value methods for IVPs

We investigate the stability properties of some classes of linear two-step methods of high order and involving derivatives of the solution, up to the order three. These methods are suitable to be used as boundary value methods (BVMs) for solving initial value problems (IVPs). The advantages with respect to other two-step BVMs are the following: (1) possibility of attaining high order of accuracy, without destroying the tridiagonality of the discretization matrix, and, (2) assurance of maintaining the accuracy of the complessive BVM of the same order as that of the basic method.

Discretizzazioni A(Θ)-stabili di equazioni differenziali astratte

We study a semi-discretization in time of linear parabolic problems by using A(Θ)-stable linear multistep methods of arbitrary order. The original problems are reduced to sequences of elliptic equations, which can be approximated by Galerkin methods. The stability and error estimates are uniform with respect to these space-discretizations.

Stiffly Stable Linear Multistep Methods of Extended Order

Certain linear k-step methods of order k (the Gear formulas) have proven very useful in the integration of stiff equations, but are of sixth order or less. In this paper, we investigate the stiff stability property of k-step methods of order $m < k$, and derive particular methods of this kind which can be viewed as extensions of the Gear formulas, and include formulas of order greater than six.