27,907 research outputs found
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
A multigrid continuation method for elliptic problems with folds
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0.
For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points
Extended Lifetime in Computational Evolution of Isolated Black Holes
Solving the 4-d Einstein equations as evolution in time requires solving
equations of two types: the four elliptic initial data (constraint) equations,
followed by the six second order evolution equations. Analytically the
constraint equations remain solved under the action of the evolution, and one
approach is to simply monitor them ({\it unconstrained} evolution).
The problem of the 3-d computational simulation of even a single isolated
vacuum black hole has proven to be remarkably difficult. Recently, we have
become aware of two publications that describe very long term evolution, at
least for single isolated black holes. An essential feature in each of these
results is {\it constraint subtraction}. Additionally, each of these approaches
is based on what we call "modern," hyperbolic formulations of the Einstein
equations. It is generally assumed, based on computational experience, that the
use of such modern formulations is essential for long-term black hole
stability. We report here on comparable lifetime results based on the much
simpler ("traditional") - formulation.
We have also carried out a series of {\it constrained} 3-d evolutions of
single isolated black holes. We find that constraint solution can produce
substantially stabilized long-term single hole evolutions. However, we have
found that for large domains, neither constraint-subtracted nor constrained
- evolutions carried out in Cartesian coordinates admit
arbitrarily long-lived simulations. The failure appears to arise from features
at the inner excision boundary; the behavior does generally improve with
resolution.Comment: 20 pages, 6 figure
Gravitational waves in dynamical spacetimes with matter content in the Fully Constrained Formulation
The Fully Constrained Formulation (FCF) of General Relativity is a novel
framework introduced as an alternative to the hyperbolic formulations
traditionally used in numerical relativity. The FCF equations form a hybrid
elliptic-hyperbolic system of equations including explicitly the constraints.
We present an implicit-explicit numerical algorithm to solve the hyperbolic
part, whereas the elliptic sector shares the form and properties with the well
known Conformally Flat Condition (CFC) approximation. We show the stability
andconvergence properties of the numerical scheme with numerical simulations of
vacuum solutions. We have performed the first numerical evolutions of the
coupled system of hydrodynamics and Einstein equations within FCF. As a proof
of principle of the viability of the formalism, we present 2D axisymmetric
simulations of an oscillating neutron star. In order to simplify the analysis
we have neglected the back-reaction of the gravitational waves into the
dynamics, which is small (<2 %) for the system considered in this work. We use
spherical coordinates grids which are well adapted for simulations of stars and
allow for extended grids that marginally reach the wave zone. We have extracted
the gravitational wave signature and compared to the Newtonian quadrupole and
hexadecapole formulae. Both extraction methods show agreement within the
numerical errors and the approximations used (~30 %).Comment: 17 pages, 9 figures, 2 tables, accepted for publication in PR
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