We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of...
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 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 are concerned with the structure of the operator
corresponding to the Lax–Friedrichs method.
At first, the phenomenae which may arise by the
naive use of the Lax–Friedrichs scheme are analyzed.
In particular, it turns out that the correct
definition of the method has to include the details
of the discretization of the initial condition
and the computational domain. Based on the results of the
discussion, we give a recipe that ensures that the
number of extrema within the discretized version...
In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL...
Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed, where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers...
The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton–Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art...
Download Results (CSV)