Showing papers in "Computational Mathematics and Mathematical Physics in 2010"

Boundary conforming Delaunay mesh generation

Abstract: A boundary conforming Delaunay mesh is a partitioning of a polyhedral domain into Delaunay simplices such that all boundary simplices satisfy the generalized Gabriel property. It’s dual is a Voronoi partition of the same domain which is preferable for Voronoi-box based finite volume schemes. For arbitrary 2D polygonal regions, such meshes can be generated in optimal time and size. For arbitrary 3D polyhedral domains, however, this problem remains a challenge. The main contribution of this paper is to show that boundary conforming Delaunay meshes for 3D polyhedral domains can be generated efficiently when the smallest input angle of the domain is bounded by arccos 1/3 ≈ 70.53°. In addition, well-shaped tetrahedra and an appropriate mesh size can be obtained. Our new results are achieved by reanalyzing a classical Delaunay refinement algorithm. Note that our theoretical guarantee on the input angle (70.53°) is still too strong for many practical situations. We further discuss variants of the algorithm to relax the input angle restriction and to improve the mesh quality.

Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium

Abstract: The paper is devoted to new applications of the ideas underlying Godunov’s method that was developed as early as in the 1950s for solving fluid dynamics problems. This paper deals with elastoplastic problems. Based on an elastic model and its modification obtained by introducing the Maxwell viscosity, a method for modeling plastic deformations is proposed.

A modification of Hutchinson’s equation

Abstract: A new mathematical object is introduced, namely, a scalar nonlinear delay differential-difference equation is considered that is a modification of Hutchinson’s equation, which is well known in ecology. The existence and stability of its relaxation self-oscillations are analyzed.

Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom

Abstract: A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended to compute complex-geometry three-dimensional flows of a diatomic gas with rotational degrees of freedom.

Simulating Flows of Incompressible and Weakly Compressible Fluids on Multicore Hybrid Computer Systems

Abstract: A logically simple algorithm based on explicit schemes for modeling flows of incompressible and weakly compressible fluids is considered. The hyperbolic variant of the quasi-gas dynamic system of equations is used as a mathematical model. An ingenious computer cluster based on NVIDIA GPUs is used for the computations.

High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations

Abstract: An approach for the creation of high-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations is proposed. New versions of up to the eighth order of accuracy inclusive are implemented. For smooth solutions, numerical experiments on a sequence of grids show that the approximate solutions produced by these versions converge to the exact one with a high order of accuracy as h → 0, where h is the maximal linear cell size of a grid. The numerical results obtained for the benchmark problem of the lid-driven cavity flow suggest that the collocations and least squares method is well suited for the numerical simulation of viscous flows.

Front motion in the parabolic reaction-diffusion problem

Abstract: A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.

Implicit numerical method for computing three-dimensional rarefied gas flows on unstructured meshes

Abstract: The method based on the numerical solution of a model kinetic equation is proposed for analyzing three-dimensional rarefied gas flows. The basic idea behind the method is the use of a second-order accurate TVD scheme on hybrid unstructured meshes in physical space and a fast implicit time discretization method without iterations at the upper level. The performance of the method is illustrated by computing test examples of three-dimensional rarefied gas flows in variously shaped channels in a wide range of Knudsen numbers.

Energy equalities and estimates for barotropic quasi-gasdynamic and quasi-hydrodynamic systems of equations

Abstract: Energy equalities are derived for the barotropic quasi-gasdynamic, modified quasi-gasdynamic, and quasi-hydrodynamic systems of equations. Global energy estimates of solutions are obtained. For the second of the systems, necessary and sufficient conditions for nonuniform and uniform Petrovskii parabolicity are derived.

Generalization of Kummer’s second theorem with applications

Abstract: The aim of this research paper is to obtain single series expression of $$ e^{ - x/2} _1 F_1 (\alpha ;2\alpha + i;x) $$ for i = 0, ±1, ±2, ±3, ±4, ±5, where 1F1(·) is the function of Kummer. For i = 0, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of $$ _2 F_1 [ - 2n,\alpha ;2\alpha + i;2]and_2 F_1 [ - 2n - 1,\alpha ;2\alpha + i;2] $$ each for i = 0, ±1, ±2, ±3, ±4, ±5 are also given. For i = 0, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressions of $$ e^{ - x} _1 F_1 (\alpha ;2\alpha + i;x) \times _1 F_1 (\alpha ;2\alpha + j;x) $$ for i, j = 0, ±1, ±2, ±3, ±4, ±5 and $$ (1 - x)^{ - a} _2 F_1 \left( {a,b,2b + j; - \tfrac{{2x}} {{1 - x}}} \right) $$ for j = 0, ±1, ±2, ±3, ±4, ±5 are given. For i = j = 0 and j = 0, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may be useful in the applicable sciences.

Unstructured tetrahedral mesh generation technology

Abstract: We present a robust unstructured tetrahedral mesh generation technology. This technology is a combination of boundary discretization methods, an advancing front technique and a Delaunay-based mesh generation technique. For boundary mesh generation we propose four different approaches using analytical boundary parameterization, interface with CAD systems, surface mesh refinement, and constructive solid geometry. These methods allow us to build a flexible grid generation technology with a user friendly interface.

Kinetic analysis of the isothermal flow in a long rectangular microchannel

Abstract: The linearized kinetic BGK model is used to study the steady Poiseuille flow of a rarefied gas in a long channel of rectangular cross section. The solution is constructed using the finite-volume method based on a TVD scheme. The basic computed characteristic is the mass flow rate through the channel. The effect of the relative width of the cross section is examined, and the difference of the solution from the one-dimensional flow between infinite parallel plates is analyzed. The numerical solution is compared to available results and to the analytical solution of the Navier-Stokes equations with no-slip and slip boundary conditions. The limits of applicability of the hydrodynamic solution are established depending on the degree of rarefaction of the flow and on the ratio of the side lengths of the channel cross section.

A method for solving the linear boundary value problem for an integro-differential equation

Abstract: A method for solving the linear boundary value problem for an integro-differential equation is proposed that is based on interval partition and the introduction of additional parameters. Necessary and sufficient conditions for the solvability of the problem are obtained.

Hessian-free metric-based mesh adaptation via geometry of interpolation error

Abstract: The article presents analysis of a new methodology for generating meshes minimizing L p -norms of the interpolation error or its gradient, p > 0. The key element of the methodology is the construction of a metric from node-based and edge-based values of a given function. For a mesh with N h triangles, we demonstrate numerically that L ∞-norm of the interpolation error is proportional to N −1 and L ∞-norm of the gradient of the interpolation error is proportional to N −1/2 . The methodology can be applied to adaptive solution of PDEs provided that edge-based a posteriori error estimates are available.

Finite-dimensional models of diffusion chaos

Abstract: Some parabolic systems of the reaction-diffusion type exhibit the phenomenon of diffusion chaos. Specifically, when the diffusivities decrease proportionally, while the other parameters of a system remain fixed, the system exhibits a chaotic attractor whose dimension increases indefinitely. Various finite-dimensional models of diffusion chaos are considered that represent chains of coupled ordinary differential equations and similar chains of discrete mappings. A numerical analysis suggests that these chains with suitably chosen parameters exhibit chaotic attractors of arbitrarily high dimensions.

Diagonally Implicit Runge-Kutta Methods for Differential Algebraic Equations of Indices Two and Three

Abstract: Diagonally implicit Runge-Kutta methods satisfying additional order conditions are examined. These conditions make it possible to solve differential algebraic equations of indices two and three to higher accuracy. Advantages of the proposed methods over other known techniques are demonstrated using test problems.

Locally one-dimensional scheme for fractional diffusion equations with robin boundary conditions

Abstract: For a fractional diffusion equation with Robin boundary conditions, locally one-dimensional difference schemes are considered and their stability and convergence are proved.

Technique for the numerical analysis of the riblet effect on temporal stability of plane flows

Abstract: Problems related to the temporal stability of laminar viscous incompressible flows in plane channels with ribbed walls are formulated, justified, and numerically solved. A new method is proposed whereby the systems of ordinary differential and algebraic equations obtained after a spatial approximation are transformed into systems of ordinary differential equations with a halved number of unknowns. New algorithms that effectively calculate stability characteristics, such as the critical Reynolds numbers, the maximum amplification of the disturbance kinetic energy density, and optimal disturbances are described and substantiated. The results of numerical experiments with riblets similar in shape to those used in practice are presented and discussed.

Algorithms for exact and approximate statistical simulation of Poisson ensembles

Abstract: Algorithms for exact and approximate statistical simulation of inhomogeneous Poisson ensembles are proposed, and their complexities are analyzed and compared. In this context, a new modification of the maximum cross section algorithm is constructed in which the sequence of rejections is determined by one value of a standard random variable.

Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-Model

Abstract: The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.

Spline interpolation on a uniform grid for functions with a boundary-layer component

Abstract: Spline interpolation of functions of one variable with a boundary-layer component is examined. Functions of this type can arise in the solution of a singularly perturbed boundary value problem on an interval. Spline interpolation formulas that are exact for the boundary-layer component are constructed, and their errors are estimated. Formulas for calculating the derivative based on the constructed interpolants are obtained. Numerical results are presented.

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

Abstract: In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.

Generalization of the hybrid monotone second-order finite difference scheme for gas dynamics equations to the case of unstructured 3D grid

Abstract: A generalization of the explicit hybrid monotone second-order finite difference scheme for the use on unstructured 3D grids is proposed. In this scheme, the components of the momentum density in the Cartesian coordinates are used as the working variables; the scheme is conservative. Numerical results obtained using an implementation of the proposed solution procedure on an unstructured 3D grid in a spherical layer in the model of the global circulation of the Titan’s (a Saturn’s moon) atmosphere are presented.

Structure of the Hessian Matrix and an Economical Implementation of Newton's Method in the Problem of Canonical Approximation of Tensors

Abstract: A tensor given by its canonical decomposition is approximated by another tensor (again, in the canonical decomposition) of fixed lower rank. For this problem, the structure of the Hessian matrix of the objective function is analyzed. It is shown that all the auxiliary matrices needed for constructing the quadratic model can be calculated so that the computational effort is a quadratic function of the tensor dimensionality (rather than a cubic function as in earlier publications). An economical version of the trust region Newton method is proposed in which the structure of the Hessian matrix is efficiently used for multiplying this matrix by vectors and for scaling the trust region. At each step, the subproblem of minimizing the quadratic model in the trust region is solved using the preconditioned conjugate gradient method, which is terminated if a negative curvature direction is detected for the Hessian matrix.

Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels

Abstract: Unsteady rarefied gas flows in narrow channels accompanied by shock wave formation and propagation were studied by solving the Boltzmann kinetic equation. The formation of a shock wave from an initial discontinuity of gas parameters, its propagation, damping, and reflection from the channel end face were analyzed. The Boltzmann equation was solved using finite differences. The collision integral was calculated on a fixed velocity grid by a conservative projection method. A detector of shock wave position was developed to keep track of the wave front. Parallel computations were implemented on a cluster of computers with the use of the MPI technology. Plots of shock wave damping and detailed flow fields are presented.

A plasmastatic model of the galathea-belt magnetic trap

Abstract: The mathematical apparatus of plasmastatics, which includes the MHD equilibrium equation and steady-state Maxwell equations, is reduced, in two-dimensional problems arising due to symmetry, to a single scalar second-order elliptic equation with a nonlinear right-hand side known as the Grad-Shafranov equation. In this paper, we numerically solve a series of boundary value problems for this equation that model equilibrium plasma configurations in the magnetic field of the belt-like galathea trap in a cylinder with two plasma embedded conductors. The mathematical model is outlined, the results of calculations of the magnetic field and plasma pressure in the cylinder depending on the parameters of the problem are presented, and the main integral characteristics of the trap are calculated. The existence and uniqueness of the solution is discussed; the limiting values of the maximal pressure at which there exists a solution of the equilibrium problem are found.

On the convergence rate of the simulated annealing algorithm

Abstract: The convergence rate of the simulated annealing algorithm is examined. It is shown that, if the objective function is nonsingular, then the number of its evaluations required to obtain the desired accuracy ɛ in the solution can be a slowly (namely, logarithmically) growing function as ɛ approaches zero.

A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

Abstract: For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O(N −2ln2 N + N 0 −1 ), where N + 1 and N 0 + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly at the rate of O(N −4ln4 N + N 0 −2 ). For fixed values of the parameter, the convergence rate is O(N −4 + N 0 −2 ).

Optimization of multiple covering of a bounded set with circles

Abstract: Numerical algorithms for the optimization of multiple covering of a bounded set G in the plane P with equal circles are proposed. The variants in which G is a connected bounded set in P or a finite set in P are considered. The circles may be centered at arbitrary points of G or at points belonging to a given set. Minimization of the radius of the given number of circles and minimization of the number of circles of a given radius are considered. Models and solution algorithms are described, and estimates of the solutions provided by most variants are given. Numerical results are presented.

Homogeneous algorithms for multiextremal optimization

Abstract: The class of homogeneous algorithms for multiextremal optimization is defined, and a number of theorems are proved, including a sufficient condition for the convergence of homogeneous algorithms to a global minimizer. An approach to the synthesis of homogeneous algorithms based on model multi-peak functions is proposed. The existing algorithms are reviewed, and a new efficient multidimensional algorithm based on the Delaunay triangulation is constructed. Some numerical results are presented.