Splitting schemes of a high degree of accuracy have now attained a very advanced stage. One modification of this method is the so-called "particles-in-cells" method: The splitting is carried out according to physical processes and is independent of the reduction in the dimension of the operators.
Splitting methods like local one-dimensional methods and hopscotch methods are often used in Western literature. Their applicability is somewhat limited, as fairly regular domains like squares are needed. Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. For the system of differential equations 1 where is a differential operator, , , the absolutely stable implicit schemes of simple approximation 2 become ineffective in the case of multi-dimensional problems.
For obtaining economical stable difference schemes methods are proposed based on the following ideas: 1 splitting of the difference schemes; 2 approximate factorization; 3 splitting weak approximation of the differential equations. In the case of equation 1 the respective difference schemes have the following form for the sake of simplicity, two fractional steps have been taken and the periodic Cauchy problem is considered : the splitting scheme: 3 the approximate factorization scheme: 4 the weak approximation scheme: 5 In the case of the schemes 3 and 4 inversion of the operator is replaced by inversion of the operator , i.
It is applicable especially to potential problems, problems of elasticity and problems of fluid dynamics. The method offers a powerful means of solving the Navier-Stokes equations and the results produced so far cover a range of Reynolds numbers far greater than that attained in earlier methods. Further development of the method should lead to complete numerical solutions of many of the boundary layer and wake problems which at present defy satisfactory treatment. As noted by the author very few applications of the method have yet been made to problems in solid mechanics and prospects for answers both in this field and other areas such as heat transfer are encouraging.
As the method is perfected it is likely to supplant traditional relaxation methods and finite element methods, especially with the increase in capability of large scale computers.
The literal translation was carried out by T. Cheron with financial support of the Northrop Corporation.
The editing of the translation was undertaken in collaboration with N. Later these results were extended by different authors to the equations of elasticity and plasticity and to multiply connected domains.
AMS :: Mathematics of Computation
Zav'yalov and Dr. Valeri L. This book is a comprehensive survey of different efficient algorithms for computing 1-D and 2-D splines. Our own main results relate to a new technique which we developed to obtain optimal error bounds for polynomial spline interpolation. This method is based on an integral representation of the error estimate. In many cases it gives minimal values of constants in error bounds for interpolation splines.
The same approach is also used to find optimal error bounds for local spline approximation methods. This book has become a standard textbook for students, researchers and engineers in Russia and over eleven thousands copies were sold. Shape-Preserving Spline Interpolation: Standard methods of spline functions do not preserve shape properties of the data.
By introducing shape control parameters into the spline structure, one can preserve various characteristics of the initial data including positivity, monotonicity, convexity, linear and planar sections. Based on interpolating splines, methods with shape control are usually called methods of shape-preserving spline interpolation. Here the main challenge is to develop algorithms that choose shape control parameters automatically. The majority of such algorithms solve the problem only for some special data.
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To solve the problem in a general setting, I gave a classification of the initial data and reduced the problem of shape-preserving interpolation to the problem of Hermite interpolation with constraints of inequality type. The solution is a C2 local generalized tension spline with additional knots. This allows for development of a local algorithm of shape-preserving spline interpolation where shape control parameters are selected automatically to meet the monotonicity and convexity constraints for the data.
Dr. Boris I. Kvasov
Its application makes it possible to give a complete solution to the shape-preserving interpolation problem for arbitrary data and isolate the sections of linearity, the angles, etc. Tension GB-Splines: In my opinion, my most significant contribution to the theory of splines, involves the development of " direct methods " for constructing explicit formulae for tension generalized basis splines GB-splines for short and finding recursive algorithms for the calculation of GB-splines.
This approach has yielded new local bases for various tension splines including among others rational, exponential, hyperbolic, variable order splines, and splines with additional knots.