887 research outputs found
A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems
We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogenous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach
Spectral/hp element methods for plane Newtonian extrudate swell
Spectral/hp element methods and an arbitrary Lagrangian-Eulerian (ALE)
moving-boundary technique are used to investigate planar Newtonian extrudate
swell. Newtonian extrudate swell arises when viscous liquids exit long die
slits. The problem is characterised by a stress singularity at the end of the
slit which is inherently difficult to capture and strongly influences the
predicted swelling of the fluid. The impact of inertia (0 <Re < 100) and slip
along the die wall on the free surface profile and the velocity and pressure
values in the domain and around the singularity are investigated. The high
order method is shown to provide high resolution of the steep pressure profile
at the singularity. The swelling ratio and exit pressure loss are compared with
existing results in the literature and the ability of high-order methods to
capture these values using significantly fewer degrees of freedom is
demonstrated
Determination of normalized electric eigenfields in microwave cavities with sharp edges
The magnetic field integral equation for axially symmetric cavities with
perfectly conducting piecewise smooth surfaces is discretized according to a
high-order convergent Fourier--Nystr\"om scheme. The resulting solver is used
to accurately determine eigenwavenumbers and normalized electric eigenfields in
the entire computational domain.Comment: 34 pages, 6 figure
Reciprocal-log approximation and planar PDE solvers
This article is about both approximation theory and the numerical solution of
partial differential equations (PDEs). First we introduce the notion of {\em
reciprocal-log} or {\em log-lightning approximation} of analytic functions with
branch point singularities at points by functions of the form , which have poles potentially distributed
along a Riemann surface. We prove that the errors of best reciprocal-log
approximations decrease exponentially with respect to and that exponential
or near-exponential convergence (i.e., at a rate ) also
holds for near-best approximations with preassigned singularities constructed
by linear least-squares fitting on the boundary. We then apply these results to
derive a "log-lightning method" for numerical solution of Laplace and related
PDEs in two-dimensional domains with corner singularities. The convergence is
near-exponential, in contrast to the root-exponential convergence for the
original lightning methods based on rational functions.Comment: 20 pages, 11 figure
Lumped finite elements for reactionâcross-diffusion systems on stationary surfaces
We consider a lumped surface finite element method (LSFEM) for the spatial approximation of reactionâdiffusion equations on closed compact surfaces in R3R3 in the presence of cross-diffusion. We provide a fully-discrete scheme by applying the ImplicitâExplicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere
On the existence of traveling waves in the 3D Boussinesq system
We extend earlier work on traveling waves in premixed flames in a
gravitationally stratified medium, subject to the Boussinesq approximation. For
three-dimensional channels not aligned with the gravity direction and under the
Dirichlet boundary conditions in the fluid velocity, it is shown that a
non-planar traveling wave, corresponding to a non-zero reaction, exists, under
an explicit condition relating the geometry of the crossection of the channel
to the magnitude of the Prandtl and Rayleigh numbers, or when the advection
term in the flow equations is neglected.Comment: 15 pages, to appear in Communications in Mathematical Physic
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