33,732 research outputs found
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
Stabilized hp-Finite Element Approximation of Partial Differential Equations with Nonnegative Characteristic Form
This paper is devoted to the a priori error analysis of the hp-version of a streamline-diffusion finite element method for partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic problems, first-order hyperbolic problems and second-order problems of mixed elliptic-parabolic-hyperbolic type. We derive error bounds which are simultaneously optimal in both the mesh size h and the spectral order p. Numerical examples are presented to confirm the theoretical results
Gaussian pulse dynamics in gain media with Kerr nonlinearity
Using the Kantorovitch method in combination with a Gaussian ansatz, we
derive the equations of motion for spatial, temporal and spatiotemporal optical
propagation in a dispersive Kerr medium with a general transverse and spectral
gain profile. By rewriting the variational equations as differential equations
for the temporal and spatial Gaussian q parameters, optical ABCD matrices for
the Kerr effect, a general transverse gain profile and nonparabolic spectral
gain filtering are obtained. Further effects can easily be taken into account
by adding the corresponding ABCD matrices. Applications include the temporal
pulse dynamics in gain fibers and the beam propagation or spatiotemporal pulse
evolution in bulk gain media. As an example, the steady-state spatiotemporal
Gaussian pulse dynamics in a Kerr-lens mode-locked laser resonator is studied
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