257 research outputs found
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Numerical Solution for Kawahara Equation by Using Spectral Methods
AbstractSome nonlinear wave equations are more difficult to investigate mathematically, as no general analytical method for their solutions exists. The Exponential Time Differencing (ETD) technique requires minimum stages to obtain the requiredaccurateness, which suggests an efficient technique relatingto computational duration thatensures remarkable stability characteristicsupon resolving nonlinear wave equations. This article solves the diagonal example of Kawahara equation via the ETD Runge-Kutta 4 technique. Implementation of this technique is proposed by short Matlab programs
Solving reaction-diffusion equations 10 times faster
The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud
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This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc
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Efficient detection of periodic orbits in high dimensional systems
This paper is concerned with developing a method for detecting unstable periodic orbits (UPOs) by stabilising transformations. Here the strategy is to transform the system of interest in such away that the orbits become stable. However, the number of such transformations becomes overwhelming as we move to higher dimensions [5, 16, 17]. We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs [1]. The real value of the set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. Here we extend this approach to high dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system - the Kuramoto-Sivashinsky equation
Extended Self Similarity works for the Burgers equation and why
Extended Self-Similarity (ESS), a procedure that remarkably extends the range
of scaling for structure functions in Navier--Stokes turbulence and thus allows
improved determination of intermittency exponents, has never been fully
explained. We show that ESS applies to Burgers turbulence at high Reynolds
numbers and we give the theoretical explanation of the numerically observed
improved scaling at both the infrared and ultraviolet end, in total a gain of
about three quarters of a decade: there is a reduction of subdominant
contributions to scaling when going from the standard structure function
representation to the ESS representation. We conjecture that a similar
situation holds for three-dimensional incompressible turbulence and suggest
ways of capturing subdominant contributions to scaling.Comment: 10 pages, 1 figure, submitted to J. Fluid Mech. (fasttrack
The Structure of Global Attractors for Dissipative Zakharov Systems with Forcing on the Torus
The Zakharov system was originally proposed to study the propagation of
Langmuir waves in an ionized plasma. In this paper, motivated by earlier work
of the first and third authors, we numerically and analytically investigate the
dynamics of the dissipative Zakharov system on the torus in 1 dimension. We
find an interesting family of stable periodic orbits and fixed points, and
explore bifurcations of those points as we take weaker and weaker dissipation.Comment: 16 pages, 7 figure
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