Richardson-extrapolated sequential splitting and its application

During numerical time integration, the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory. In this case one usually reduces the step size and repeats the computation, while the results obtained for the coarser grid are not used. However, we can also combine the two solutions and obtain a better result. This idea is based on the Richardson extrapolation, a general technique for increasing the order of an approximation method. This technique also allows us to estimate the absolute error of the underlying method. In this paper we apply Richardson extrapolation to the sequential splitting, and investigate the performance of the resulting scheme on several test examples.     

Effects of emulsifier charge and concentration on pancreatic lipolysis. 1. In the absence of bile salts

An in vitro study is performed with sunflower oil-in-water emulsions to clarify the effects of type of used emulsifier, its concentration, and reaction time on the degree of oil lipolysis, α. Anionic, nonionic, and cationic surfactants are studied as emulsifiers. For all systems, three regions are observed when surfactant concentration is scaled with the critical micelle concentration, C(S)/cmc: (1) At C(S) < cmc, α ≈ 0.5 after 30 min and increases up to 0.9 after 4 h. (2) At C(S) ≈ 3 × cmc, α ≈ 0.15 after 30 min and increases steeply up to 0.9 after 2 h for the cationic and nonionic surfactants, whereas it remains around 0.2 for the anionic surfactants. (3) At C(S) above certain threshold value, α = 0 for all studied surfactants, for reaction time up to 8 h. Additional experiments show that the lipase hydrolyzes molecularly soluble substrate (tributirin) at C(S) > cmc, which is a proof that these surfactants do not denature or block the enzyme active center. Thus, we conclude that the mechanism of enzyme inhibition by these surfactants is the formation of a dense adsorption layer on an oil drop surface, which displaces the lipase from direct contact with the triglycerides.     

On the scattering of torsional elastic waves from axisymmetric defects in coated pipes

Long range ultrasonic testing is now a well established method for examining in-service degradation in pipelines. In order to protect pipelines from the surrounding environment it is common for viscoelastic coatings to be applied to the outer surface. These coatings are, however, known to impact on the ability of long range ultrasonic techniques to locate degradation, or defects, within a coated pipe. The coating dissipates sound energy travelling along the pipe, attenuating both the incident and reflected signals making responses from defects difficult to detect. This article aims to investigate the influence of a viscoelastic coating on the ability of long range ultrasonic testing to detect a defect in an axisymmetric pipe. The article focuses on understanding the behaviour of the fundamental torsional mode and quantifying the effect of bitumen coatings on reflection coefficients generated by axisymmetric defects. Reflection coefficients are measured experimentally for coated and uncoated pipes and compared to theoretical predictions generated using numerical mode matching and a hybrid finite element technique. Good agreement between prediction and measurement is observed for uncoated pipes, and it is shown that the theoretical methods presented here are fast and efficient making them suitable for studying long pipe runs. However, when studying coated pipes agreement between theory and prediction is observed to be poor for predictions based on those bulk acoustic properties currently reported in the literature for bitumen. Good agreement is observed only after conducting a parametric study to identify more appropriate values for the bulk acoustic properties. Furthermore, the reflection coefficients obtained for the fundamental torsional mode in a coated pipe show that significant sound attenuation is present over relatively short lengths of coating, thus quantifying those problems commonly encountered with the use of long range ultrasonic testing on coated pipes in the field.     

General scheme for solving linear algebraic problems by direct methods

Assume that A is an m × n real matrix with m?n and rank(A)=n. Let b be a real vector with m components and consider the problem of finding $\text{x=A}{}^2\text{b}$, where $\text{A}{}^2\text{=(A}{}^{\mathrm{T}}\text{A)}{}^{\mathrm{?1}}\text{A}{}^{\mathrm{T}}$. A general scheme for solving this problem is described. Many well-known and commonly used in practice direct methods can be found as special cases within the general scheme. This is illustrated by four examples. The general scheme can be used to study the common properties of the direct methods. The usefulness of this approach in the efforts to improve the efficiency of the direct methods for sparse matrices is demonstrated by formulating an algorithm that can be applied to any particular method belonging to the general scheme. This algorithm is implemented in several subroutines solving linear algebraic problems by different direct methods. Numerical results, obtained in a wide range of runs with these subroutines, are given.     

On the additive splitting procedures and their computer realization

Two additive splitting procedures are defined and studied in this paper. It is shown that these splitting procedures have good stability properties. Some other splitting procedures, which are traditionally used in mathematical models used in many scientific and engineering fields, are sketched. All splitting procedures are tested by using six different numerical methods for solving differential equations. Many conclusions, which are related both to the comparison of the additive splitting procedures with the other splitting procedures and to the influence of the numerical methods for solving differential equations on the accuracy of the splitting procedures, are drawn.     

Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important to select fast, robust and reliable methods for their solution, also in the case where fast modern computers are available. Since the coefficient matrices of the systems are normally sparse (i.e. most of their elements are zeros), the first requirement is to efficiently exploit the sparsity. However, this is normally not sufficient when the systems are very large. The computation of preconditioners based on approximate LU-factorizations and their use in the efforts to increase further the efficiency of the calculations will be discussed in this paper. Computational experiments based on comprehensive comparisons of many numerical results that are obtained by using ten well-known methods for solving systems of linear algebraic equations (the direct Gaussian elimination and nine iterative methods) will be reported. Most of the considered methods are preconditioned Krylov subspace algorithms.     

Richardson Extrapolation combined with the sequential splitting procedure and the <Emphasis Type="Italic">θ</Emphasis>-method

Initial value problems for systems of ordinary differential equations (ODEs) are solved numerically by using a combination of (a) the θ-method, (b) the sequential splitting procedure and (c) Richardson Extrapolation. Stability results for the combined numerical method are proved. It is shown, by using numerical experiments, that if the combined numerical method is stable, then it behaves as a second-order method.