14,463 research outputs found
A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations
The Douglas--Rachford and Peaceman--Rachford splitting methods are common
choices for temporal discretizations of evolution equations. In this paper we
combine these methods with spatial discretizations fulfilling some easily
verifiable criteria. In the setting of linear dissipative evolution equations
we prove optimal convergence orders, simultaneously in time and space. We apply
our abstract results to dimension splitting of a 2D diffusion problem, where a
finite element method is used for spatial discretization. To conclude, the
convergence results are illustrated with numerical experiments
Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation
We consider Kirchhoff equations with a small parameter epsilon in front of
the second-order time-derivative, and a dissipative term whose coefficient may
tend to 0 as t -> + infinity (weak dissipation).
In this note we present some recent results concerning existence of global
solutions, and their asymptotic behavior both as t -> + infinity and as epsilon
-> 0. Since the limit equation is of parabolic type, this is usually referred
to as a hyperbolic-parabolic singular perturbation problem.
We show in particular that the equation exhibits hyperbolic or parabolic
behavior depending on the values of the parameters.Comment: 20 pages, 2 tables, 1 figure, conference paper (7th ISAAC congress,
London 2009
The discrete energy method in numerical relativity: Towards long-term stability
The energy method can be used to identify well-posed initial boundary value
problems for quasi-linear, symmetric hyperbolic partial differential equations
with maximally dissipative boundary conditions. A similar analysis of the
discrete system can be used to construct stable finite difference equations for
these problems at the linear level. In this paper we apply these techniques to
some test problems commonly used in numerical relativity and observe that while
we obtain convergent schemes, fast growing modes, or ``artificial
instabilities,'' contaminate the solution. We find that these growing modes can
partially arise from the lack of a Leibnitz rule for discrete derivatives and
discuss ways to limit this spurious growth.Comment: 18 pages, 22 figure
Operator splitting for dissipative delay equations
We investigate Lie-Trotter product formulae for abstract nonlinear evolution
equations with delay. Using results from the theory of nonlinear contraction
semigroups in Hilbert spaces, we explain the convergence of the splitting
procedure. The order of convergence is also investigated in detail, and some
numerical illustrations are presented.Comment: to appear in Semigroup Foru
Smoothed Particle Hydrodynamics and Magnetohydrodynamics
This paper presents an overview and introduction to Smoothed Particle
Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we
give a basic grounding in the fundamentals of SPH, showing how the equations of
motion and energy can be self-consistently derived from the density estimate.
We then show how to interpret these equations using the basic SPH interpolation
formulae and highlight the subtle difference in approach between SPH and other
particle methods. In doing so, we also critique several `urban myths' regarding
SPH, in particular the idea that one can simply increase the `neighbour number'
more slowly than the total number of particles in order to obtain convergence.
We also discuss the origin of numerical instabilities such as the pairing and
tensile instabilities. Finally, we give practical advice on how to resolve
three of the main issues with SPMHD: removing the tensile instability,
formulating dissipative terms for MHD shocks and enforcing the divergence
constraint on the particles, and we give the current status of developments in
this area. Accompanying the paper is the first public release of the NDSPMHD
SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD
algorithms that can be used to test many of the ideas and used to run all of
the numerical examples contained in the paper.Comment: 44 pages, 14 figures, accepted to special edition of J. Comp. Phys.
on "Computational Plasma Physics". The ndspmhd code is available for download
from http://users.monash.edu.au/~dprice/ndspmhd
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