Multi-dimensional Boundary Value Problems Research Articles

Приближенное определение потерь характеристической скорости и приращения высоты полета верхних ступеней многоступенчатых ракет-носителей

An important role is assigned to design-ballistic calculations at the initial stage of developing a new type of the launch vehicle. Results of such calculations are a necessary condition in solving a more general multidimensional boundary value problem of the ballistic design. It should be noted that ballistic conditions determine the design-ballistic parameters ratio to solve the boundary value problem, and its algorithms could be used as components of the higher-level optimization problems. In order to save computer time, development of the so-called fast algorithms implemented using the analytical approaches to solving a ballistic problem becomes expedient. In the logical chain of calculation dependencies making part of the ballistic calculation algorithms, the main are those relations that connect losses in the characteristic velocity with the design and ballistic parameters. They are understood as the minimum set of parameters that uniquely determine the rocket trajectory. Analytical calculated dependences of the characteristic velocity losses and of the flight altitude increments on the design-ballistic parameters of the second and third stages of three-stage launch vehicles were derived.

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral‐type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self‐adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral‐type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.

The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.

On the minimal residual methods for solving diffusion-convection SLAEs

The objective of this research is to develop and to study iterative methods in the Krylov subspaces for solving systems of linear algebraic equations (SLAEs) with non-symmetric sparse matrices of high orders arising in the approximation of multi-dimensional boundary value problems on the unstructured grids. These methods are also relevant in many applications, including diffusion-convection equations. The considered algorithms are based on constructing ATA — orthogonal direction vectors calculated using short recursions and providing global minimization of a residual at each iteration. Methods based on the Lanczos orthogonalization, AT — preconditioned conjugate residuals algorithm, as well as the left Gauss transform for the original SLAEs are implemented. In addition, the efficiency of these iterative processes is investigated when solving algebraic preconditioned systems using an approximate factorization of the original matrix in the Eisenstat modification. The results of a set of computational experiments for various grids and values of convective coefficients are presented, which demonstrate a sufficiently high efficiency of the approaches under consideration.

Two-Level Iterative Methods for Solving the Saddle Point Problems

Iterative processes in the Krylov subspaces for solving large ill conditioned saddle-type SLAEs with sparse matrices arising in finite difference, finite volume, and finite element approximations of multidimensional boundary value problems with complex geometric and functional properties of the initial data, characteristic of many relevant applications are studied. Combined two-level iterative algorithms using efficient Chebyshev acceleration and variational the conjugate directions methods, as well as the Golub-Kahan bi-diagonalization algorithms in the Krylov subspaces are considered. Examples of two-dimensional and three-dimensional filtration problems are used to study the resource consumption and computational performance of the proposed algorithms, as well as their scalable parallization on the multiprocessor systems with distributed and hierarchical shared memory.

Nested Alternating - Triangular Incomplete Factorization Methods

We consider several versions of incomplete nested factorization methods for solving the large systems of linear algebraic equations (SLAEs) with sparse matrices which arise in grid approximations of the multi-dimensional boundary value problems. Our approach is based on the two-level iterative process in the Krylov subspaces in 3D case. Corresponding hierarchical incomplete factorization is applied to the block tridiagonal matrix structure. At the upper level, the diagonal blocks correspond to 2D grid subproblems which are factorized in the line-by-line framework. Instead of the low and upper triangular matrix factors, the alternating triangular matrices are used, which allows to apply the parallel counter sweeping approaches. The improvement of preconditioners is made by means of generalized compensation principles. To solve SLAE iterative conjugate direction methods in Krylov subspaces are applied. The efficiency of the proposed methods are demonstrated on the set of representative test problems.

Numerical Study on Single and Multi-Dimensional Boundary Value Problems by the Method of Weighted Residual

The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.

О вопросах распараллеливания крыловских итерационных методов

Mathematical questions of various computational technologies of parallelelized iterative processes of Krylov’s type for solving large sparse symmetric and non-symmetric SLAEs, obtained in grid approximations of multi-dimensional boundary value problems for PDEs, are considered. Example are presented by finite approximations in gas-hydrodynamical applications, where five unknowns in each node are defined and corresponding SLAEs have small-block structure. The base of used algorithms is flexible generalized minimal residual, FGMRES, method with dynamical preconditioners of additive type, which presents an upper level of two-step iterarive Swartz algorithm. High performance of algebraic solvers is provided by using different approaches: domain decompositions of various topologies, boundary conditions and sizes of subdomain overlapping, coarse grid correction, deflation and aggregation, and incomplete factorizations of matrices. The unified formulations of using algorithms as well as the questions of computational efficiency and scalable parallelization at the geterogenous supercomputers are described. The examples of technical requirements for peculiarities of program implementation of the libraries of parallel algorithms for solving systems of linear algebraic equation, are presented.

Функциональность и технологии алгебраических решателей в библиотеке Krylov

Article describes functional capabilities and software implementation peculiarities of parallel algorithms library Krylov, which is oriented on the solution of large systems of linear algebraic equations with sparse symmetric and unsymmetric matrices (positive definite and semi-definite) obtained from discrete approximations of multidimensional boundary value problems for partial differential equations on unstructured meshes. The library includes two-level iterative methods in Krylov subspaces; preconditioning of the latter is based on the balanced decomposition of the computational domain with variable sizes of subdomain overlapping and different boundary conditions on interfacing boundaries. Program implementations use typical compressed sparse matrix data formats. Results of numerical experiments are presented which demonstrate the efficiency of parallelization for typical ill-conditioned problems.

Методы параллельного решения СЛАУ на системах с распределенной памятью в библиотеке Krylov

The paper presents an approach to creation of black-box parallel iterative solver, which is used in Krylov library for solving systems of linear algebraic equations (SLAEs) with large sparse matrices in CSR format arising from discretization of multidimensional boundary value problems. A variant of one-dimensional algebraic decomposition method is offered. The algorithm is based on breadth-first search of SLAE’s adjacency graph that allows to reduce the matrix to block-tridiagonal form. The algebraic solver is based on additive Schwarz method which naturally suits distributed memory computer systems. The generalized minimal residual method is used to solve the SLAEs arising from relations on subdomains’ boundaries. Auxiliary subdomain systems are solved with Intel MKL’s multithreaded direct solver PARDISO. Implemented algorithms were tested on the numerical solution of the series of computational mathematics problems, such as problems of hydrodynamics, diffusion-convection equations, problems of electromagnetism and others. Adduced numerical experiments results show the effectiveness of the presented algorithms for multiprocessor computational systems with distributed memory.

Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type

A new definition of Lie invariance for nonlinear multi-dimensional boundary value problems (BVPs) is proposed by the generalization of known definitions to much wider classes of BVPs. The class of (1+3)-dimensional nonlinear BVPs of the Stefan type, modeling the process of melting and evaporation of metals, is studied in detail. Using the definition proposed, the group classification problem for this class of BVPs is solved and some reductions (with physical meaning) to BVPs of lower dimensionality are made. Examples of how to construct exact solutions of the (1+3)-dimensional nonlinear BVP with the correctly specified coefficients are presented.

A multi-dimensional shooting algorithm for the two-facility location–allocation problem with dense demand

We develop an efficient allocation-based solution framework for a class of two-facility location–allocation problems with dense demand data. By formulating the problem as a multi-dimensional boundary value problem, we show that previous results for the discrete demand case can be extended to problems with highly dense demand data. Further, this approach can be generalized to non-convex allocation decisions. This formulation is illustrated for the Euclidean metric case by representing the affine bisector with two points. A specialized multi-dimensional shooting algorithm is presented and illustrated on an example. Comparisons with two alternative methods through a computational study confirm the efficiency of the proposed methodology.

Time-optimal steering of a point mass onto the surface of a sphere at zero velocity

The problem of the time-optimal steering of a point mass onto the surface of a sphere at zero velocity, by a control force of bounded magnitude is investigated. It is assumed that the surface is penetrable and that the point may “land” on the sphere either from the outside or from the inside. An optimal control, in the open-loop and feedback form of trajectories the optimal time and the Bellman function are constructed using Pontrya'gin's maximum principle. The multidimensional boundary-value problem is reduced, by introducing self-similar variables, to the numerical solution of an algebraic equation of degree four and a transcendental equation. It is shown that the boundary-value problem degenerates when the optimal trajectory is nearly linear; a solution of the synthesis problem is constructed in the degenerate case. The efficacy of the approach proposed here is illustrated by specific examples in which families of trajectories are computed, and by an analysis of control regimes.

Reducing the boundary value problems of elasticity theory for cracked bodies to boundary integral equations

A general method has been proposed for constructing integral representations of general solutions and boundary integral equations of multidimensional boundary value problems of mathematical physics for regions with cuts. It involves the use of the theory of generalized functions, and in particular of the surface delta function. At first, the boundary value problems of Dirichlet and Neumann were studied for n-dimensional Poisson and Helmholtz equations in a space with cuts along piecewise-smooth surfaces. After that the method is extended to the case of a system of differential equations. In this way the basic spatial and plane problems of elasticity theory were considered for an anisotropic infinite body with cracks under static and dynamic loading. The corresponding axisymmetric problems were also studied.

A method of characteristics for the numerical solution of multi-dimensional boundary value problems

A method of characteristics for the numerical solution of multi-dimensional boundary value problems

On the difference-variational method for multidimensional boundary value problems

On the difference-variational method for multidimensional boundary value problems