Theoretical Analysis of Fully Conservative Difference Schemes with Adaptive Viscosity

For the equations of gas dynamics in Eulerian variables, a family of two-layer in time completely conservative difference schemes with space-profiled time weights is constructed. Considerable attention is paid to the methods of constructing regularized flows of mass, momentum, and internal energy that do not violate the properties of complete conservatism of difference schemes of this class, to the analysis of their amplitudes and the possibility of their use on non-uniform grids. Effective preservation of the balance of internal energy in this type of divergent difference schemes is ensured by the absence of constantly operating sources of difference origin that produce "computational"entropy (including those based on singular features of the solution). The developed schemes can be easily generalized in order to calculate high-temperature flows in media that are nonequilibrium in temperature (for example, in a plasma with a difference in the temperatures of the electronic and ionic components), when, with the set of variables necessary for describing the flow, it is not enough to equalize the total energy balance.

Economical factorized schemes for third-order pseudoparabolic equations

Изучены экономичные факторизованные схемы для псевдопараболических уравнений третьего порядка. На основе общей теории устойчивости разностных схем доказаны устойчивость и сходимость разностных схем. Economical factorized schemes for pseudo-parabolic equations of the third order are studied. On the basis of the general theory of stability of difference schemes, the stability and convergence of difference schemes are proved.

On the efficiency of high-order difference schemes for the Schro¨dinger equation

Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties

On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x=f(x), found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Coopers theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Coopers theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as t0.Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.