Solution of boundary value and eigenvalue problems for second order elliptic operators in the plane using pseudoanalytic formal powers. (arXiv:1002.1110v1 [math.AP])

Solution of boundary value and eigenvalue problems for second order elliptic operators in the plane using pseudoanalytic formal powers. (arXiv:1002.1110v1 [math.AP]): "

We propose a method for solving boundary value and eigenvalue problems for
the elliptic operator D=divpgrad+q in the plane using pseudoanalytic function
theory and in particular pseudoanalytic formal powers. Under certain conditions
on the coefficients p and q with the aid of pseudoanalytic function theory a
complete system of null solutions of the operator can be constructed following
a simple algorithm consisting in recursive integration. This system of
solutions is used for solving boundary value and spectral problems for the
operator D in bounded simply connected domains. We study theoretical and
numerical aspects of the method.

"

Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times

Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times: "

Abstract Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space
are analyzed. It is shown that along both the solution of the nonlinear Schrödinger equation as well as the solution of the
semi-discretized equation the actions of the linear Schrödinger equation are approximately conserved over long times. This
also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation
over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary
spatial dimension.

• Content Type Journal Article
• DOI 10.1007/s10208-010-9059-z
• Authors
  
  
  • Ludwig Gauckler, Universität Tübingen Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tübingen Germany
  • Christian Lubich, Universität Tübingen Mathematisches Institut Auf der Morgenstelle 10 D-72076 Tübingen Germany
  
  

• Journal Foundations of Computational Mathematics
• Online ISSN 1615-3383
• Print ISSN 1615-3375

"

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order: "

Abstract The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations
of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be
the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth
and a Hölder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous
operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.

• Content Type Journal Article
• DOI 10.1007/s10208-010-9061-5
• Authors
  
  
  • Etienne Emmrich, Universität Bielefeld Fakultät für Mathematik Postfach 100131 33501 Bielefeld Germany
  • Mechthild Thalhammer, Leopold-Franzens-Universität Innsbruck Institut für Mathematik Technikerstraße 13/VII 6020 Innsbruck Austria
  
  

• Journal Foundations of Computational Mathematics
• Online ISSN 1615-3383
• Print ISSN 1615-3375

"

On stability of numerical schemes via frozen coefficients and the magnetic induction equations

On stability of numerical schemes via frozen coefficients and the magnetic induction equations: "

Abstract We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations
in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term)
finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle
of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant
coefficient discretizations are stable.

We apply this general result to the special case of magnetic induction equations and show that high order SBP-SAT schemes
are energy stable even with boundary closures. Furthermore, we introduce a modified discretization of lower order terms and
show that the discrete divergence of this scheme is bounded. The discrete divergence is shown to converge to zero under certain
assumptions. Computations supporting our theoretical results are also presented.

• Content Type Journal Article
• DOI 10.1007/s10543-010-0249-5
• Authors
  
  
  • Siddhartha Mishra, University of Oslo Centre of Mathematics for Applications P.B. 1053 Blindern 0316 Oslo Norway
  • Magnus Svärd, University of Oslo Centre of Mathematics for Applications P.B. 1053 Blindern 0316 Oslo Norway
  
  

• Journal BIT Numerical Mathematics
• Online ISSN 1572-9125
• Print ISSN 0006-3835

"

Practical Quasi-Newton algorithms for singular nonlinear systems

Practical Quasi-Newton algorithms for singular nonlinear systems: "

Abstract Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause
a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose
two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong
to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep
iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.

• Content Type Journal Article
• Category Original Paper
• DOI 10.1007/s11075-010-9367-z
• Authors
  
  
  • Sandra Buhmiler, University of Novi Sad Department of Mathematics, Faculty of Engineering 21000 Novi Sad Serbia
  • Nataša Krejić, University of Novi Sad Department of Mathematics and Informatics, Faculty of Science Trg Dositeja Obradovića 4 21000 Novi Sad Serbia
  • Zorana Lužanin, University of Novi Sad Department of Mathematics and Informatics, Faculty of Science Trg Dositeja Obradovića 4 21000 Novi Sad Serbia
  
  

• Journal Numerical Algorithms
• Online ISSN 1572-9265
• Print ISSN 1017-1398

"