10 research outputs found
Low regularity integrators for semilinear parabolic equations with maximum bound principles
This paper is concerned with conditionally structure-preserving, low
regularity time integration methods for a class of semilinear parabolic
equations of Allen-Cahn type. Important properties of such equations include
maximum bound principle (MBP) and energy dissipation law; for the former, that
means the absolute value of the solution is pointwisely bounded for all the
time by some constant imposed by appropriate initial and boundary conditions.
The model equation is first discretized in space by the central finite
difference, then by iteratively using Duhamel's formula, first- and
second-order low regularity integrators (LRIs) are constructed for time
discretization of the semi-discrete system. The proposed LRI schemes are proved
to preserve the MBP and the energy stability in the discrete sense.
Furthermore, their temporal error estimates are also successfully derived under
a low regularity requirement that the exact solution of the semi-discrete
problem is only assumed to be continuous in time. Numerical results show that
the proposed LRI schemes are more accurate and have better convergence rates
than classic exponential time differencing schemes, especially when the
interfacial parameter approaches zero.Comment: 24 page
A Second Order Fully-discrete Linear Energy Stable Scheme for a Binary Compressible Viscous Fluid Model
We present a linear, second order fully discrete numerical scheme on a
staggered grid for a thermodynamically consistent hydrodynamic phase field
model of binary compressible fluid flow mixtures derived from the generalized
Onsager Principle. The hydrodynamic model not only possesses the variational
structure, but also warrants the mass, linear momentum conservation as well as
energy dissipation. We first reformulate the model in an equivalent form using
the energy quadratization method and then discretize the reformulated model to
obtain a semi-discrete partial differential equation system using the
Crank-Nicolson method in time. The numerical scheme so derived preserves the
mass conservation and energy dissipation law at the semi-discrete level. Then,
we discretize the semi-discrete PDE system on a staggered grid in space to
arrive at a fully discrete scheme using the 2nd order finite difference method,
which respects a discrete energy dissipation law. We prove the unique
solvability of the linear system resulting from the fully discrete scheme. Mesh
refinements and two numerical examples on phase separation due to the spinodal
decomposition in two polymeric fluids and interface evolution in the gas-liquid
mixture are presented to show the convergence property and the usefulness of
the new scheme in applications
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described