8,818 research outputs found
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Introduction: Localized Structures in Dissipative Media: From Optics to Plant Ecology
Localised structures in dissipative appears in various fields of natural
science such as biology, chemistry, plant ecology, optics and laser physics.
The proposed theme issue is to gather specialists from various fields of
non-linear science toward a cross-fertilisation among active areas of research.
This is a cross-disciplinary area of research dominated by the nonlinear optics
due to potential applications for all-optical control of light, optical
storage, and information processing. This theme issue contains contributions
from 18 active groups involved in localized structures field and have all made
significant contributions in recent years.Comment: 14 pages, 0 figure, submitted to Phi. Trasaction Royal Societ
On Validating an Astrophysical Simulation Code
We present a case study of validating an astrophysical simulation code. Our
study focuses on validating FLASH, a parallel, adaptive-mesh hydrodynamics code
for studying the compressible, reactive flows found in many astrophysical
environments. We describe the astrophysics problems of interest and the
challenges associated with simulating these problems. We describe methodology
and discuss solutions to difficulties encountered in verification and
validation. We describe verification tests regularly administered to the code,
present the results of new verification tests, and outline a method for testing
general equations of state. We present the results of two validation tests in
which we compared simulations to experimental data. The first is of a
laser-driven shock propagating through a multi-layer target, a configuration
subject to both Rayleigh-Taylor and Richtmyer-Meshkov instabilities. The second
test is a classic Rayleigh-Taylor instability, where a heavy fluid is supported
against the force of gravity by a light fluid. Our simulations of the
multi-layer target experiments showed good agreement with the experimental
results, but our simulations of the Rayleigh-Taylor instability did not agree
well with the experimental results. We discuss our findings and present results
of additional simulations undertaken to further investigate the Rayleigh-Taylor
instability.Comment: 76 pages, 26 figures (3 color), Accepted for publication in the ApJ
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