166 research outputs found
Extension of Chebfun to Periodic Functions
Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted
The chebop system for automatic solution of differential equations
In MATLAB, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, based on the previously developed chebfun system in object-oriented MATLAB. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution
Computing with functions in spherical and polar geometries I. The sphere
A collection of algorithms is described for numerically computing with smooth
functions defined on the unit sphere. Functions are approximated to essentially
machine precision by using a structure-preserving iterative variant of Gaussian
elimination together with the double Fourier sphere method. We show that this
procedure allows for stable differentiation, reduces the oversampling of
functions near the poles, and converges for certain analytic functions.
Operations such as function evaluation, differentiation, and integration are
particularly efficient and can be computed by essentially one-dimensional
algorithms. A highlight is an optimal complexity direct solver for Poisson's
equation on the sphere using a spectral method. Without parallelization, we
solve Poisson's equation with million degrees of freedom in one minute on
a standard laptop. Numerical results are presented throughout. In a companion
paper (part II) we extend the ideas presented here to computing with functions
on the disk.Comment: 23 page
Computing planar and spherical choreographies
An algorithm is presented for numerical computation of choreographies in the
plane in a Newtonian potential and on the sphere in a cotangent potential. It
is based on stereographic projection, approximation by trigonometric
polynomials, and quasi-Newton and Newton optimization methods with exact
gradient and exact Hessian matrix. New choreographies on the sphere are
presented
Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
This paper discusses a connection between scalar convex conservation laws and
Pontryagin's minimum principle. For flux functions for which an associated
optimal control problem can be found, a minimum value solution of the
conservation law is proposed. For scalar space-independent convex conservation
laws such a control problem exists and the minimum value solution of the
conservation law is equivalent to the entropy solution. This can be seen as a
generalization of the Lax--Oleinik formula to convex (not necessarily uniformly
convex) flux functions. Using Pontryagin's minimum principle, an algorithm for
finding the minimum value solution pointwise of scalar convex conservation laws
is given. Numerical examples of approximating the solution of both
space-dependent and space-independent conservation laws are provided to
demonstrate the accuracy and applicability of the proposed algorithm.
Furthermore, a MATLAB routine using Chebfun is provided (along with
demonstration code on how to use it) to approximately solve scalar convex
conservation laws with space-independent flux functions
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
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