21,319 research outputs found
Preconditioned fully implicit PDE solvers for monument conservation
Mathematical models for the description, in a quantitative way, of the
damages induced on the monuments by the action of specific pollutants are often
systems of nonlinear, possibly degenerate, parabolic equations. Although some
the asymptotic properties of the solutions are known, for a short window of
time, one needs a numerical approximation scheme in order to have a
quantitative forecast at any time of interest. In this paper a fully implicit
numerical method is proposed, analyzed and numerically tested for parabolic
equations of porous media type and on a systems of two PDEs that models the
sulfation of marble in monuments. Due to the nonlinear nature of the underlying
mathematical model, the use of a fixed point scheme is required and every step
implies the solution of large, locally structured, linear systems. A special
effort is devoted to the spectral analysis of the relevant matrices and to the
design of appropriate iterative or multi-iterative solvers, with special
attention to preconditioned Krylov methods and to multigrid procedures.
Numerical experiments for the validation of the analysis complement this
contribution.Comment: 26 pages, 13 figure
Gradient-based estimation of Manning's friction coefficient from noisy data
We study the numerical recovery of Manning's roughness coefficient for the
diffusive wave approximation of the shallow water equation. We describe a
conjugate gradient method for the numerical inversion. Numerical results for
one-dimensional model are presented to illustrate the feasibility of the
approach. Also we provide a proof of the differentiability of the weak form
with respect to the coefficient as well as the continuity and boundedness of
the linearized operator under reasonable assumptions using the maximal
parabolic regularity theory.Comment: 19 pages, 3 figure
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