8,785 research outputs found
Semi-Lagrangian methods for parabolic problems in divergence form
Semi-Lagrangian methods have traditionally been developed in the framework of
hyperbolic equations, but several extensions of the Semi-Lagrangian approach to
diffusion and advection--diffusion problems have been proposed recently. These
extensions are mostly based on probabilistic arguments and share the common
feature of treating second-order operators in trace form, which makes them
unsuitable for mass conservative models like the classical formulations of
turbulent diffusion employed in computational fluid dynamics. We propose here
some basic ideas for treating second-order operators in divergence form. A
general framework for constructing consistent schemes in one space dimension is
presented, and a specific case of nonconservative discretization is discussed
in detail and analysed. Finally, an extension to (possibly nonlinear) problems
in an arbitrary number of dimensions is proposed. Although the resulting
discretization approach is only of first order in time, numerical results in a
number of test cases highlight the advantages of these methods for applications
to computational fluid dynamics and their superiority over to more standard low
order time discretization approaches
On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity
The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used
in numerical relativity for the solution of both hyperbolic and parabolic
partial differential equations. We here extend the recent work on the stability
of this scheme for hyperbolic equations by investigating the properties when
the average between the predicted and corrected values is made with unequal
weights and when the scheme is applied to a parabolic equation. We also propose
a variant of the scheme in which the coefficients in the averages are swapped
between two corrections leading to systematically larger amplification factors
and to a smaller numerical dispersion.Comment: 7 pages, 3 figure
Rich: Open Source Hydrodynamic Simulation on a Moving Voronoi Mesh
We present here RICH, a state of the art 2D hydrodynamic code based on
Godunov's method, on an unstructured moving mesh (the acronym stands for Racah
Institute Computational Hydrodynamics). This code is largely based on the code
AREPO. It differs from AREPO in the interpolation and time advancement scheme
as well as a novel parallelization scheme based on Voronoi tessellation. Using
our code we study the pros and cons of a moving mesh (in comparison to a static
mesh). We also compare its accuracy to other codes. Specifically, we show that
our implementation of external sources and time advancement scheme is more
accurate and robust than AREPO's, when the mesh is allowed to move. We
performed a parameter study of the cell rounding mechanism (Llyod iterations)
and it effects. We find that in most cases a moving mesh gives better results
than a static mesh, but it is not universally true. In the case where matter
moves in one way, and a sound wave is traveling in the other way (such that
relative to the grid the wave is not moving) a static mesh gives better results
than a moving mesh. Moreover, we show that Voronoi based moving mesh schemes
suffer from an error, that is resolution independent, due to inconsistencies
between the flux calculation and change in the area of a cell. Our code is
publicly available as open source and designed in an object oriented, user
friendly way that facilitates incorporation of new algorithms and physical
processes
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
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