22,882 research outputs found
A moving mesh method with variable relaxation time
We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.Comment: 21 page
Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems
We present a modification to the Berger and Oliger adaptive mesh refinement
algorithm designed to solve systems of coupled, non-linear, hyperbolic and
elliptic partial differential equations. Such systems typically arise during
constrained evolution of the field equations of general relativity. The novel
aspect of this algorithm is a technique of "extrapolation and delayed solution"
used to deal with the non-local nature of the solution of the elliptic
equations, driven by dynamical sources, within the usual Berger and Oliger
time-stepping framework. We show empirical results demonstrating the
effectiveness of this technique in axisymmetric gravitational collapse
simulations. We also describe several other details of the code, including
truncation error estimation using a self-shadow hierarchy, and the
refinement-boundary interpolation operators that are used to help suppress
spurious high-frequency solution components ("noise").Comment: 31 pages, 15 figures; replaced with published versio
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
Monolithic simulation of convection-coupled phase-change - verification and reproducibility
Phase interfaces in melting and solidification processes are strongly
affected by the presence of convection in the liquid. One way of modeling their
transient evolution is to couple an incompressible flow model to an energy
balance in enthalpy formulation. Two strong nonlinearities arise, which account
for the viscosity variation between phases and the latent heat of fusion at the
phase interface.
The resulting coupled system of PDE's can be solved by a single-domain
semi-phase-field, variable viscosity, finite element method with monolithic
system coupling and global Newton linearization. A robust computational model
for realistic phase-change regimes furthermore requires a flexible
implementation based on sophisticated mesh adaptivity. In this article, we
present first steps towards implementing such a computational model into a
simulation tool which we call Phaseflow.
Phaseflow utilizes the finite element software FEniCS, which includes a
dual-weighted residual method for goal-oriented adaptive mesh refinement.
Phaseflow is an open-source, dimension-independent implementation that, upon an
appropriate parameter choice, reduces to classical benchmark situations
including the lid-driven cavity and the Stefan problem. We present and discuss
numerical results for these, an octadecane PCM convection-coupled melting
benchmark, and a preliminary 3D convection-coupled melting example,
demonstrating the flexible implementation. Though being preliminary, the latter
is, to our knowledge, the first published 3D result for this method. In our
work, we especially emphasize reproducibility and provide an easy-to-use
portable software container using Docker.Comment: 20 pages, 8 figure
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
We propose a novel second order in time numerical scheme for
Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme
is based on second order convex-splitting for the Cahn-Hilliard equation and
pressure-projection for the Navier-Stokes equation. We show that the scheme is
mass-conservative, satisfies a modified energy law and is therefore
unconditionally stable. Moreover, we prove that the scheme is uncondition- ally
uniquely solvable at each time step by exploring the monotonicity associated
with the scheme. Thanks to the weak coupling of the scheme, we design an
efficient Picard iteration procedure to further decouple the computation of
Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by
the mixed finite element method. Ample numerical experiments are performed to
validate the accuracy and efficiency of the numerical scheme
A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations
This paper presents a new resolution strategy for multi-scale streamer
discharge simulations based on a second order time adaptive integration and
space adaptive multiresolution. A classical fluid model is used to describe
plasma discharges, considering drift-diffusion equations and the computation of
electric field. The proposed numerical method provides a time-space accuracy
control of the solution, and thus, an effective accurate resolution independent
of the fastest physical time scale. An important improvement of the
computational efficiency is achieved whenever the required time steps go beyond
standard stability constraints associated with mesh size or source time scales
for the resolution of the drift-diffusion equations, whereas the stability
constraint related to the dielectric relaxation time scale is respected but
with a second order precision. Numerical illustrations show that the strategy
can be efficiently applied to simulate the propagation of highly nonlinear
ionizing waves as streamer discharges, as well as highly multi-scale nanosecond
repetitively pulsed discharges, describing consistently a broad spectrum of
space and time scales as well as different physical scenarios for consecutive
discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for
several month stay of Z. Bonaventura at Laboratory EM2C as visiting
Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS)
for providing the basis of the multiresolution kernel of MR CHORUS, code
developed for compressible Navier-Stokes equations (D\'eclaration d'Invention
DI 03760-01). Accepted for publication; Journal of Computational Physics
(2011) 1-2
Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology
In standard models of cardiac electrophysiology, including the bidomain and
monodomain models, local perturbations can propagate at infinite speed. We
address this unrealistic property by developing a hyperbolic bidomain model
that is based on a generalization of Ohm's law with a Cattaneo-type model for
the fluxes. Further, we obtain a hyperbolic monodomain model in the case that
the intracellular and extracellular conductivity tensors have the same
anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is
equivalent to a cable model that includes axial inductances, and the relaxation
times of the Cattaneo fluxes are strictly related to these inductances. A
purely linear analysis shows that the inductances are negligible, but models of
cardiac electrophysiology are highly nonlinear, and linear predictions may not
capture the fully nonlinear dynamics. In fact, contrary to the linear analysis,
we show that for simple nonlinear ionic models, an increase in conduction
velocity is obtained for small and moderate values of the relaxation time. A
similar behavior is also demonstrated with biophysically detailed ionic models.
Using the Fenton-Karma model along with a low-order finite element spatial
discretization, we numerically analyze differences between the standard
monodomain model and the hyperbolic monodomain model. In a simple benchmark
test, we show that the propagation of the action potential is strongly
influenced by the alignment of the fibers with respect to the mesh in both the
parabolic and hyperbolic models when using relatively coarse spatial
discretizations. Accurate predictions of the conduction velocity require
computational mesh spacings on the order of a single cardiac cell. We also
compare the two formulations in the case of spiral break up and atrial
fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure
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