1,726 research outputs found
Convergence of a semi-discretization scheme for the Hamilton--Jacobi equation: a new approach with the adjoint method
We consider a numerical scheme for the one dimensional time dependent Hamilton--Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L. C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the O(sqrt{h}) convergence rate in terms of the L^infty norm and O(h) in terms of the L^1 norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper
Continuous dependence estimates for nonlinear fractional convection-diffusion equations
We develop a general framework for finding error estimates for
convection-diffusion equations with nonlocal, nonlinear, and possibly
degenerate diffusion terms. The equations are nonlocal because they involve
fractional diffusion operators that are generators of pure jump Levy processes
(e.g. the fractional Laplacian). As an application, we derive continuous
dependence estimates on the nonlinearities and on the Levy measure of the
diffusion term. Estimates of the rates of convergence for general nonlinear
nonlocal vanishing viscosity approximations of scalar conservation laws then
follow as a corollary. Our results both cover, and extend to new equations, a
large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link
with the results in [51,59
Compactness estimates for Hamilton-Jacobi equations depending on space
We study quantitative estimates of compactness in
for the map , that associates to every given initial data the corresponding solution of a
Hamilton-Jacobi equation with a convex and coercive Hamiltonian
. We provide upper and lower bounds of order on the
the Kolmogorov -entropy in of the image through
the map of sets of bounded, compactly supported initial data.
Quantitative estimates of compactness, as suggested by P.D. Lax, could provide
a measure of the order of "resolution" and of "complexity" of a numerical
method implemented for this equation. We establish these estimates deriving
accurate a-priori bounds on the Lipschitz, semiconcavity and semiconvexity
constant of a viscosity solution when the initial data is semiconvex. The
derivation of a small time controllability result is also fundamental to
establish the lower bounds on the -entropy.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1403.455
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