342 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems
For conservative mechanical systems, the so-called Caughey series are known to define the class of damping matrices that preserve eigenspaces. In particular, for finite-dimensional systems, these matrices prove to be a polynomial of one reduced matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values.
This paper first recasts this result in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators). Second, generalizations to the infinite-dimensional case are considered. They consists of extending the previous polynomial class to rational functions and more general
functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian
Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains
In this paper, we propose a fast spectral-Galerkin method for solving PDEs
involving integral fractional Laplacian in , which is built upon
two essential components: (i) the Dunford-Taylor formulation of the fractional
Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions
(MCFs) as basis functions. As a result, the fractional Laplacian can be fully
diagonalised, and the complexity of solving an elliptic fractional PDE is
quasi-optimal, i.e., with being the number of modes in
each spatial direction. Ample numerical tests for various decaying exact
solutions show that the convergence of the fast solver perfectly matches the
order of theoretical error estimates. With a suitable time-discretization, the
fast solver can be directly applied to a large class of nonlinear fractional
PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation
by using the fourth-order time-splitting method together with the proposed
MCF-spectral-Galerkin method.Comment: This article has a total of 24 pages and including 22 figure
A unified meshfree pseudospectral method for solving both classical and fractional PDEs
In this paper, we propose a meshfree method based on the Gaussian radial
basis function (RBF) to solve both classical and fractional PDEs. The proposed
method takes advantage of the analytical Laplacian of Gaussian functions so as
to accommodate the discretization of the classical and fractional Laplacian in
a single framework and avoid the large computational cost for numerical
evaluation of the fractional derivatives. These important merits distinguish it
from other numerical methods for fractional PDEs. Moreover, our method is
simple and easy to handle complex geometry and local refinement, and its
computer program implementation remains the same for any dimension .
Extensive numerical experiments are provided to study the performance of our
method in both approximating the Dirichlet Laplace operators and solving PDE
problems. Compared to the recently proposed Wendland RBF method, our method
exactly incorporates the Dirichlet boundary conditions into the scheme and is
free of the Gibbs phenomenon as observed in the literature. Our studies suggest
that to obtain good accuracy the shape parameter cannot be too small or too
big, and the optimal shape parameter might depend on the RBF center points and
the solution properties.Comment: 24 pages; 15 figure
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