Search results for "Numerical approximation"
showing 6 items of 16 documents
The average over a sphere
1980
Abstract The N points ri and the N segments ΔΩi of the unit sphere used in the numerical approximation of the average over the sphere are optimized to approximate the average of the set of spherical harmonics {;Yl,m;l = 0, 1, 2, …, L}; up to L = 18. The symmetry of f( r ) can be taken into acount by using only a distinct subquantity of the N point {; r i , ΔΩ i }; . Sets for N = 48n (n = 1, 2, …, 6) are tabulated. The advantage of the method is shown by the calculation of a powder Mossbauer spectrum including electric and magnetic hyperfine interactions.
Indirect Methods for Optimal Control Problems
2003
This chapter is dedicated to the numerical approximation of Optimal Control Problems. The algorithms are based on the necessary conditions for optimality which allow us to use a descent method for the minimization of the cost functional.
A brief overview on the numerical behavior of an implicit meshless method and an outlook to future challenges
2015
In this paper recent results on a leapfrog ADI meshless formulation are reported and some future challenges are addressed. The method benefits from the elimination of the meshing task from the pre-processing stage in space and it is unconditionally stable in time. Further improvements come from the ease of implementation, which makes computer codes very flexible in contrast to mesh based solver ones. The method requires only nodes at scattered locations and a function and its derivatives are approximated by means of a kernel representation. A perceived obstacle in the implicit formulation is in the second order differentiations which sometimes are eccesively sensitive to the node configurat…
Numerical Approximation of Elliptic Variational Problems
2003
This chapter is dedicated to the study of Elliptic Variational Inequalities (EVI). Different forms of such an EVI are considered. The Ritz—Galerkin discretization method is introduced, and methods to approximate the solution of an EVI are presented. The finite dimensional subspaces are built by use of the Finite Element Method. The discretized problems are solved using variants of the Successive OverRelaxation (SOR) method. The algorithms are tested on a typical example. The way to develop computer programs is carefully analysed.
Fronts propagating with signal dependent speed in limited diffusion and related Hamilton-Jacobi formulations
2021
We consider a class of limited diffusion equations and explore the formation of diffusion fronts as the result of a combination of diffusive and hyperbolic transport. We analyze a new class of Hamilton-Jacobi equations arising from the convective part of general Fokker-Planck equations ruled by a non-negative diffusion coefficient that depends on the unknown and on the gradient of the unknown. We explore the main features of the solution of the Hamilton-Jacobi equations that contain shocks and propose a suitable numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker-Planck equation. We analyze three model problems covering d…
Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in Cartesian coordinates
2018
With a Hamilton–Jacobi equation in Cartesian coordinates as a starting point, it is common to use a system of ordinary differential equations describing the continuation of first-order derivatives of phase-space perturbations along a reference ray. Such derivatives can be exploited for calculating geometrical spreading on the reference ray and for establishing a framework for second-order extrapolation of traveltime to points outside the reference ray. The continuation of first-order derivatives of phase-space perturbations has historically been referred to as dynamic ray tracing. The reason for this is its importance in the process of calculating amplitudes along the reference ray. We exte…