Håvard Berland

Bio: Håvard Berland is an academic researcher from Equinor. The author has contributed to research in topics: Exponential integrator & Exponential function. The author has an hindex of 7, co-authored 9 publications receiving 358 citations. Previous affiliations of Håvard Berland include Norwegian University of Science and Technology.

EXPINT---A MATLAB package for exponential integrators

Abstract: Recently, a great deal of attention has been focused on the construction of exponential integrators for semilinear problems. In this article we describe a MATLAB1 package which aims to facilitate the quick deployment and testing of exponential integrators, of Runge--Kutta, multistep, and general linear type. A large number of integrators are included in this package along with several well-known examples. The so-called φ functions and their evaluation is crucial for accuracy, stability, and efficiency of exponential integrators, and the approach taken here is through a modification of the scaling and squaring technique, the most common approach used for computing the matrix exponential.

B-series and Order Conditions for Exponential Integrators

Abstract: We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of $B$-series and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [{\em {Math. Comp.,}} 46 (1986), pp. 119--133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.

Solving the nonlinear Schrödinger equation using exponential integrators

Abstract: This PhD-thesis contains an introduction and six research papers sorted chronologically, of which the first four are accepted for publication. The introduction aims at giving a very brief summary of the background theory needed for the following papers. Also, some motivation of the issues addressed by the papers is given. Paper I discusses algebraic structures of ordered rooted trees and their applications to Lie group integrators. Results from Hopf algebra theory on elementary differentials for Lie group integrators are used, and applications to order analysis and backward error analysis are given. Paper II, III, IV, and V are primarily on exponential integrators, a class of numerical schemes tailored the solution of stiff systems of systems of ordinary differential equations. Paper II discusses classical order analysis and gives some theoretical results on the form of the integrators, applicable for the construction of high order exponential integrators. Paper III is on an implementation of exponential integrators on computers, and source code, available electronically, accompanies the paper. Paper IV includes an analytical and numerical study of the performance of two classes of exponential integrators on the nonlinear Schrodinger equation. Paper V is a numerical study of behaviour over long integration invervals on the nonlinear Schrodinger equation, using nonlinear spectral theory for determining validity of the numerical solution and thereby jugdging the numerical integrators. At last, in Paper VI, properties of a class of exponential like functions, essential in exponential integrators, are derived, using an approach based on Lie group theory.

Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Hopf π- Algebras

Abstract: This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems

Abstract: The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction.

The deal.II Library, Version 9.0

Abstract: Abstract This paper provides an overview of the new features of the finite element library deal.II version 9.0.

A review of exponential integrators for first order semi-linear problems

Abstract: . (1.1)have emerged as a viable alternative. In [88] it is claimed that the largest per-formance improvement in the solution of partial differential equations will comefrom better time integration technology. In the early fifties, the phenomenon ofstiffness was first discovered by Curtis and Hirschfelder [18]. Stiffness effectivelyyields explicit integrators useless, as stability rather than accuracy governs howthe integrator performs. It could be said that more integrators have been de-veloped to overcome the phenomenon of stiffness, than any other property thata differential equation may have. Stiffness was also the reason for the introduc-tion of exponential integrators. The recent wave of publications on exponentialintegrators, have mainly been concerned with the time integration of spatiallydiscretized parabolic and hyperbolic partial differential equations.Various methods have been developed for the differential equation (1.1). Thefirst paper to construct what are now known as exponential integrators, was byCertaine [15], published in 1960. This paper constructed two exponential inte-grators based on the Adams–Moulton methods of order two and three. Thesemethods are members of the Exponential Time Differencing (ETD) methods,which find approximations to the integral in the variation of constants formula,using an algebraic polynomial approximation to the nonlinear term. Other pa-pers on this subject include [5, 16, 44, 43, 58, 77]. In 1967, Lawson published[62], which provided a novel approach to solving stiff problems. Integratorswere constructed, which solve exactly the linear part of the problem and thenused a change of variables to cast the problem in a form, which a traditionalexplicit method can be used to solve the transformed equation, the approxi-mate solution is then back transformed. These methods are commonly knownas Integrating Factor (IF) methods. Further papers on this subject include1