Bojan Orel

Parallel Runge-Kutta methods with real eigenvalues

Abstract This paper describes a parallel implementation of implicit Runge–Kutta methods with real eigenvalues. The construction of collocation-type methods as well as of methods with the stability function below the main diagonal of the Pade tableau is described. A method for estimating the local error and an implementation of these methods on parallel machines are proposed.

Complexity Theory for Lie-Group Solvers

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge?Kutta?Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusion is that for order four, six, and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Lie-group method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g., Krylov-subspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods.

The growth mechanism of zinc oxide and hydrozincite: a study using electron microscopies and in situ SAXS

The growth mechanism of ZnO is investigated by a combination of electron microscopy and in situ small-angle X-ray scattering (SAXS). The particles are prepared by the precipitation of zinc nitrate with urea. Depending on the reaction conditions, ZnO, hydrozincite, or a mixture of both phases is detected in our system. The molecular precursors and complexation reactions of the formation process are numerically predicted by the partial-charge model. The condensation and complexation reactions lead to the formation of nanoparticle building units up to a size of 10 nm. Afterwards, the nanoparticles immediately self-assemble into micro-sized particles.

Real pole approximations to the exponential function

Rational approximations to the exponential function with real, not necessarily distinct poles are studied in this paper. The orthogonality relation is established in order to show that the zeros of the collocation polynomial of the corresponding Runge-Kutta method are all real, simple and positive. It is proven, that approximants with the smallest error constant are the Restricted Padé approximants of Nørsett. Some results concerning acceptability properties are given.

Computations with half-range Chebyshev polynomials

An efficient construction of two non-classical families of orthogonal polynomials is presented in the paper. The so-called half-range Chebyshev polynomials of the first and second kinds were first introduced by Huybrechs in Huybrechs (2010) [5]. Some properties of these polynomials are also shown. Every integrable function can be represented as an infinite series of sines and cosines of these polynomials, the so-called half-range Chebyshev-Fourier (HCF) series. The second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated HCF series, where two matrices are constructed for this purpose: the differentiation and the multiplication matrix.

Extrapolated Magnus Methods

This paper describes the use of extrapolation with Magnus methods for the solution of a system of linear differential equations. The idea is a generalization of extrapolation with symmetric methods for the numerical solution of ODEs, where each extrapolation step increases the order of the method by 2.

Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems

A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval , we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF) series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010) and further analyzed by Orel and Perne (2012). The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation number for the new class of Chebyshev-Fourier-collocation (CFC) methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC) methods.

GISAXS view of induced morphological changes in nanostructured CeVO4 thin films

Nanostructured CeVO4 films, designed for applications in electrochemical cells and electrochromic devices, were obtained on glass substrates by the sol-gel process. An analysis of morphological modifications in these films, induced by ultrasonication, annealing, and introduction of lithium ions, was performed, using the grazing-incidence small-angle X-ray scattering technique (GISAXS). The GISAXS results are discussed and related with complementary examinations of the same films in real space, performed by scanning electron microscopy on a different length scale.

Coarse-Grain Parallelisation of Multi-Implicit Runge-Kutta Methods

A parallel implementation for a multi-implicit Runge-Kutta method (MIRK) with real eigenvalues is decribed. The parallel method is analysed and the algorithm is devised. For the problem with d domains, the amount of work within the s-stage MIRK method, associated with the solution of system, is proportional to (sd)3, in contrast to the simple implicit finite difference method (IFD) where the amount of work is proportional to d3. However, it is shown that s-stage MIRK admits much greater time steps for the same order of error. Additionally, the proposed parallelisation transforms the system of the dimension sd to s independent sub-systems of dimension d. The amount of work for the sequential solution of such systems is proportional to sd3. The described parallel algorithm enables the solving of each of the s subsystems on a separate processor; finally, the amount of work is again d3, but the profit of a larger time step still remains. To test the theory, a comparative example of the 3-D heat transfer in a human heart with 643 domains is shown and numerically calculated by 3-stage MIRK.