3,019 research outputs found
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
Interactions of inert confiners with explosives
The deformation of an inert confiner by a steady detonation wave in an
adjacent explosive is investigated for cases where the confiner is suciently strong
(or the explosive suciently weak) such that the overall change in the sound speed
of the inert is small. A coupling condition which relates the pressure to the deflection
angle along the explosive-inert interface is determined. This includes its dependence
on the thickness of the inert, for cases where the initial sound speed of the inert
is less than or greater than the detonation speed in the explosive (supersonic and
subsonic inert
ows, respectively). The deformation of the inert is then solved by
prescribing the pressure along the interface. In the supersonic case, the detonation
drives a shock into the inert, subsequent to which the
ow in the inert consists
of alternating regions of compression and tension. In this case reverberations or
`ringing' occurs along both the deflected interface and outer edge of the inert. For
the subsonic case, the
flow in the interior of the inert is smooth and shockless.
The detonation in the explosive initially defl
ects the smooth interface towards the
explosive. For sufficiently thick inerts in such cases, it appears that the deflection
of the confiner would either drive the detonation speed in the explosive up to the
sound speed of the inert or drive a precursor wave ahead of the detonation in the
explosive. Transonic cases, where the inert sound speed is close to the detonation
speed, are also considered. It is shown that the confinement affect of the inert on
the detonation is enhanced as sonic conditions are approached from either side
A multiscale method for heterogeneous bulk-surface coupling
In this paper, we construct and analyze a multiscale (finite element) method
for parabolic problems with heterogeneous dynamic boundary conditions. As
origin, we consider a reformulation of the system in order to decouple the
discretization of bulk and surface dynamics. This allows us to combine
multiscale methods on the boundary with standard Lagrangian schemes in the
interior. We prove convergence and quantify explicit rates for low-regularity
solutions, independent of the oscillatory behavior of the heterogeneities. As a
result, coarse discretization parameters, which do not resolve the fine scales,
can be considered. The theoretical findings are justified by a number of
numerical experiments including dynamic boundary conditions with random
diffusion coefficients
Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect
We investigate a non-isothermal diffuse-interface model that describes the
dynamics of two-phase incompressible flows with thermo-induced Marangoni
effect. The governing PDE system consists of the Navier--Stokes equations
coupled with convective phase-field and energy transport equations, in which
the surface tension, fluid viscosity and thermal diffusivity are temperature
dependent functions. First, we establish the existence and uniqueness of local
strong solutions when the spatial dimension is two and three. Then in the two
dimensional case, assuming that the -norm of the initial temperature
is suitably bounded with respect to the coefficients of the system, we prove
the existence of global weak solutions as well as the existence and uniqueness
of global strong solutions.Comment: accepted by Euro. J. Appl. Math. in June 201
Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions
We analytically and numerically analyze groundwater flow in a homogeneous
soil described by the Richards equation, coupled to surface water represented
by a set of ordinary differential equations (ODE's) on parts of the domain
boundary, and with nonlinear outflow conditions of Signorini's type. The
coupling of the partial differential equation (PDE) and the ODE's is given by
nonlinear Robin boundary conditions. This article provides two major new
contributions regarding these infiltration conditions. First, an existence
result for the continuous coupled problem is established with the help of a
regularization technique. Second, we analyze and validate a solver-friendly
discretization of the coupled problem based on an implicit-explicit time
discretization and on finite elements in space. The discretized PDE leads to
convex spatial minimization problems which can be solved efficiently by
monotone multigrid. Numerical experiments are provided using the DUNE numerics
framework.Comment: 34 pages, 5 figure
- …