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We discuss the occurrence of oscillations
when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and
non-staggered second order Nessyahu-Tadmor (NT) schemes.
Although these schemes are monotone or TVD, respectively,
oscillations may be introduced at local data extrema.
The dependence of oscillatory properties on the numerical viscosity
coefficient is investigated rigorously for the LFt schemes, illuminating also
the properties of Rusanov's method. It turns...
We consider the system of partial differential equations governing
the one-dimensional flow of two superposed immiscible layers of
shallow water. The difficulty in this system comes
from the coupling terms involving some derivatives of the unknowns
that make the system nonconservative, and eventually nonhyperbolic.
Due to these terms, a numerical scheme obtained by performing an
arbitrary scheme to each layer, and using time-splitting or
other similar techniques leads to instabilities in...
We describe a numerical method for the equation in with Dirichlet boundary and initial conditions which is a combination of the method of characteristics and the finite-difference method. We prove both an a priori local error-estimate of a high order and stability. Example 3.3 indicates that our approximate solutions are disturbed only by a minimal amount of the artificial diffusion.
An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.
In this paper we investigate a mixed parabolic-hyperbolic initial boundary value problem in two disconnected intervals with Robin-Dirichlet conjugation conditions. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate is obtained.
We consider an asymptotic preserving numerical scheme initially proposed by F. Filbet and S. Jin [J. Comput. Phys. 229 (2010)] and G. Dimarco and L. Pareschi [SIAM J. Numer. Anal. 49 (2011) 2057–2077] in the context of nonlinear and stiff kinetic equations. Here, we propose a convergence analysis of such a scheme for the approximation of a system of transport equations with a nonlinear source term, for which the asymptotic limit is given by a conservation law. We investigate the convergence of the...
We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group.
The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme,
we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like where h is the mesh step. Such...
The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed....
We consider in all curvature equation
where G is a nondecreasing function and
curv(u) is the curvature of the level
line passing by x. These equations are invariant with respect
to any contrast change u → g(u), with g nondecreasing.
Consider the contrast invariant operator
.
A Matheron theorem asserts that all contrast invariant operator
T can be put in a form
.
We show the asymptotic equivalence of both formulations.
More precisely, we show that all curvature equations can be obtained...
This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.
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