47,317 research outputs found
CREATION AND EVALUATION OF THE STRUCTURES GRID IN CURVILINEAR AREAS
The article concerns methods of a structural curvilinear grid constructing in areas of geometrically complex shape and its evaluation from the quality point of view. Equidistribution methods based on differential equations were used to construct the grid at the boundary and inside the region. The numerical solution of differential equations was realized by the finite difference method. For the problems of uniform arrangement of grid nodes on the boundary and for the problems of constructing curved grids inside the region, implicit difference schemes were constructed and methods of scalar sweep and alternating directions were used. The results of numerical calculations are obtained and graphs of curved grids are presented for different numbers of grid nodes. The quality of the grid was studied according to four criteria such as orthogonality, elongation, convexity and adaptability, which corresponds to the division of the considered area into equal subdomains, i.e. cells
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
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