499 research outputs found
Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation
We are concerned with the convergence of spectral method for the numerical
solution of the initial-boundary value problem associated to the Korteweg-de
Vries-Kawahara equation (in short Kawahara equation), which is a transport
equation perturbed by dispersive terms of 3rd and 5th order. This equation
appears in several fluid dynamics problems. It describes the evolution of small
but finite amplitude long waves in various problems in fluid dynamics. These
equations are discretized in space by the standard Fourier- Galerkin spectral
method and in time by the explicit leap-frog scheme. For the resulting fully
discrete, conditionally stable scheme we prove an L2-error bound of spectral
accuracy in space and of second-order accuracy in time.Comment: 15 page
Operator splitting for the Benjamin-Ono equation
In this paper we analyze operator splitting for the Benjamin-Ono equation,
u_t = uu_x + Hu_xx, where H denotes the Hilbert transform. If the initial data
are sufficiently regular, we show the convergence of both Godunov and Strang
splitting.Comment: 18 Page
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
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