9,200 research outputs found
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally
challenging task, especially when a large set of eigenvalues is computed, or
just when particularly large eigenvalues are sought. This is a consequence of
the highly oscillatory behaviour of the solutions corresponding to high
eigenvalues, which forces a naive integrator to take increasingly smaller
steps. We will discuss some techniques that yield uniform approximation over
the whole eigenvalue spectrum and can take large steps even for high
eigenvalues. In particular, we will focus on methods based on coefficient
approximation which replace the coefficient functions of the Sturm-Liouville
problem by simpler approximations and then solve the approximating problem. The
use of (modified) Magnus or Neumann integrators allows to extend the
coefficient approximation idea to higher order methods
Applying numerical continuation to the parameter dependence of solutions of the Schr\"odinger equation
In molecular reactions at the microscopic level the appearance of resonances
has an important influence on the reactivity. It is important to predict when a
bound state transitions into a resonance and how these transitions depend on
various system parameters such as internuclear distances. The dynamics of such
systems are described by the time-independent Schr\"odinger equation and the
resonances are modeled by poles of the S-matrix. Using numerical continuation
methods and bifurcation theory, techniques which find their roots in the study
of dynamical systems, we are able to develop efficient and robust methods to
study the transitions of bound states into resonances. By applying Keller's
Pseudo-Arclength continuation, we can minimize the numerical complexity of our
algorithm. As continuation methods generally assume smooth and well-behaving
functions and the S-matrix is neither, special care has been taken to ensure
accurate results. We have successfully applied our approach in a number of
model problems involving the radial Schr\"odinger equation
Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations
Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation
Hardness Results for Structured Linear Systems
We show that if the nearly-linear time solvers for Laplacian matrices and
their generalizations can be extended to solve just slightly larger families of
linear systems, then they can be used to quickly solve all systems of linear
equations over the reals. This result can be viewed either positively or
negatively: either we will develop nearly-linear time algorithms for solving
all systems of linear equations over the reals, or progress on the families we
can solve in nearly-linear time will soon halt
Phase diagram of the chromatic polynomial on a torus
We study the zero-temperature partition function of the Potts antiferromagnet
(i.e., the chromatic polynomial) on a torus using a transfer-matrix approach.
We consider square- and triangular-lattice strips with fixed width L, arbitrary
length N, and fully periodic boundary conditions. On the mathematical side, we
obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the
square and triangular lattices. On the physical side, we obtain the exact
``phase diagrams'' for these strips of width L and infinite length, and from
these results we extract useful information about the infinite-volume phase
diagram of this model: in particular, the number and position of the different
phases.Comment: 72 pages (LaTeX2e). Includes tex file, three sty files, and 26
Postscript figures. Also included are Mathematica files transfer6_sq.m and
transfer6_tri.m. Final version to appear in Nucl. Phys.
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