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Vol 11, No 3 (2018)
- Year: 2018
- Articles: 7
- URL: https://journals.rcsi.science/1995-4239/issue/view/12389
Article
Analysis of Numerical Differentiation Formulas in a Boundary Layer on a Shishkin Grid
Abstract
A problem of numerical differentiation of functions with large gradients in a boundary layer is investigated. The problem is that for functions with large gradients and a uniform grid the relative error of the classical difference formulas for derivatives may be considerable. It is proposed to use a Shishkin grid to obtain a relative error of the formulas that is independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform with respect to the small parameter. In the case of a uniform grid, a boundary layer region is indicated outside of which the numerical differentiation formulas have an error that is uniform with respect to the small parameter. The results of numerical experiments are presented.
193-203
On the Congruent Selection of Jordan Blocks from a Singular Square Matrix
Abstract
The concept of a regularizing decomposition was introduced by R. Horn and V. Sergeichuk. It means the representation of a square matrix by a direct sum of Jordan blocks with zero on the principal diagonal and a nonsingular matrix. Such a representation is attained via congruence transformations and differs from the Jordan normal form. For the reasons explained in this paper, we prefer to speak of an SR decomposition (in other words, a singular-regular decomposition) of a matrix rather than a regularizing decomposition. Accordingly, algorithms providing this decomposition are called SR algorithms.We develop a rational algorithm that considerably simplifies the SR algorithms proposed by Horn and Sergeichuk.
204-207
A New Class of Exact Solutions to the Two-Dimensional Eikonal Equation Where the Velocity in the Medium Depends on One of the Spatial Coordinates Alone
Abstract
A new method to obtain exact solutions to the two-dimensional eikonal equation where the velocity of the medium depends on one of the spatial coordinates alone is proposed. Several examples of reducing the initial problem to one or several ordinary differential equations by substituting the solution into a suitable general form are presented. The dynamics of wave propagation is illustrated for each of the solutions thus obtained.
208-219
Comparison of Radial Basis Functions
Abstract
A survey of algorithms for approximation of multivariate functions with radial basis function (RBF) splines is presented. Algorithms of interpolating, smoothing, selecting the smoothing parameter, and regression with splines are described in detail. These algorithms are based on the feature of conditional positive definiteness of the spline radial basis function. Several families of radial basis functions generated by means of conditionally completely monotone functions are considered. Recommendations for the selection of the spline basis and preparation of initial data for approximation with the help of the RBF spline are given.
220-235
Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation
Abstract
An inverse boundary value problem for a nonlinear parabolic equation is considered. Two-sided estimates for the norms of values of a nonlinear operator in terms of those of a corresponding linear operator are obtained.On this basis, two-sided estimates for the modulus of continuity of a nonlinear inverse problem in terms of that of a corresponding linear problem are obtained. A method of auxiliary boundary conditions is used to construct stable approximate solutions to the nonlinear inverse problem. An accurate (to an order) error estimate for the method of auxiliary boundary conditions is obtained on a uniform regularization class.
236-255
Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids
Abstract
Some boundary conditions used to numerically simulate tsunami generation and propagation are studied. Special attention is given to generating boundary conditions thatmake it possible to simulate tsunami waves with desired characteristics (amplitude, time period and, in general, waveform). Since the water flow velocity in a propagating tsunami wave is uniquely defined by its height and ocean depth, one can simulate a wave propagating from the boundary into the simulation area. This can be done by specifying the wave height and water flow velocity on the boundary. This method is used to numerically simulate the propagation of a tsunami from the source to the coast on a sequence of refined grids. In this numerical experiment the wave parameters are transferred from the larger area to the subarea via boundary conditions. This method can also generate a wave that has certain characteristics on a specified line.
256-267
Mixed Methods for Optimal Control Problems
Abstract
In this paper, we investigate a posteriori error estimates of amixed finite elementmethod for elliptic optimal control problems with an integral constraint. The gradient for ourmethod belongs to the square integrable space instead of the classical H(div; Ω) space. The state and co-state are approximated by the P02-P1 (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. Using duality argument method and energy method, we derive the residual a posteriori error estimates for all variables.
268-277