A numerical method for a system of equations with a small parameter and a point source on an infinite interval

A method for solving a boundary-value problem on an infinite interval is considered for a linear system of second-order ordinary differential equations with a small parameter at the highest derivatives and a point source. The question is addressed of reduction of this problem to a finite interval. A mesh, condensing in the boundary layer, is used for numerical solution of a system of singularly perturbed equations on a finite interval.     

Nonlinear data-bounded polynomial approximations and their applications in ENO methods

A class of high-order data-bounded polynomials on general meshes are derived and analyzed in the context of numerical solutions of hyperbolic equations. Such polynomials make it possible to circumvent the problem of Runge-type oscillations by adaptively varying the stencil and order used, but at the cost of only enforcing C ⁰ solution continuity at data points. It is shown that the use of these polynomials, based on extending the work of Berzins (SIAM Rev 1(4):624–627, 2007) to nonuniform meshes, provides a way to develop positivity preserving polynomial approximations of potentially high order for hyperbolic equations. The central idea is to use ENO (Essentially Non Oscillatory) type approximations but to enforce additional restrictions on how the polynomial order is increased. The question of how high a polynomial order should be used will be considered, with respect to typical numerical examples. The results show that this approach is successful but that it is necessary to provide sufficient resolution inside a front if high-order methods of this type are to be used, thus emphasizing the need to consider nonuniform meshes.     

Optimal m-stage Runge-Kutta schemes for steady-state solutions of hyperbolic equations

A numerical scheme is developed to find optimal parameters and time step of m-stage Runge-Kutta (RK) schemes for accelerating the convergence to -steady-state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m-stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.     

A splitting algorithm for the wavelet transform of cubic splines on a nonuniform grid

For cubic splines with nonuniform nodes, splitting with respect to the even and odd nodes is used to obtain a wavelet expansion algorithm in the form of the solution to a three-diagonal system of linear algebraic equations for the coefficients. Computations by hand are used to investigate the application of this algorithm for numerical differentiation. The results are illustrated by solving a prediction problem.     

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.     

Numerical analysis of the dispersion of acoustoelectric waves in a layer with a thickness axis of symmetry for its physical propertes

A numerical method is developed for studying normal electroelastic waves in a layer of piezoelectric materials with mm2 rhombic symmetry, and a second order thickness atris of symmetry. The main system of equations is reduced to eight hamiltonian-type equations in the thickness coordinate. For harmonic waves, the generalized spectral problem is solved numerically taking into account the even (odd) character of the solution with respect to the central plane of the layer. Some solutions of specific problems are analyzed. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 162–169, 1999.     

具有非光滑边界层函数的线性变系数抛物型方程初边值问题的差分格式

本文利用非均匀网格和指数型拟合差分方法给出了具有非光滑边界层函数的线性抛物型方程关于小参数ε一致收敛的差分格式.文章还给出了误差估计和数值结果.     

Transient piezothermoelastic analysis of a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator

In this study, the theoretical treatment of transient piezothermoelastic problem is developed for a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator due to nonuniform heat supply. By using the exact solutions for cross-ply laminate and piezoelectric layer of crystal class mm2, the theoretical analysis of a transient piezothermoelasticity is developed for a simple supported cylindrical composite panel under the state of plane strain. Analysis of a piezothermoelastic problem leads to an appropriate electric potential applied to the piezoelectric layer which suppresses the induced thermoelastic displacement in the radial direction at the midpoint on the free surface of the cross-ply laminate. Some numerical results for the temperature change, the displacement, the stress in a transient state when the transient thermoelastic displacement is controlled are shown in figures.     

Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities

In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution.     

Equations for the vibrations of a plate with general anisotropy

The exact form of the displacement field for an elastic layer with general anisotropy and nonuniform thickness is constructed. The low-frequency part of the solution, which satisfies the equations for the vibrations of a plate, is found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 122–135, 1987.     

The method of fundamental solutions for linear diffusion-reaction equations

The paper presents an extension of the solution procedure based on the method of fundamental solutions proposed earlier in the literature for solving linear diffusion reaction equations in nonregular geometries in two and three dimensions. The solution procedure utilizes the fundamental solution to the problem along with boundary collocation to result in a grid-free numerical scheme. A new heuristic for source location is utilized along with orthogonal collocation in nonsmooth domains to improve the accuracy of the solution. The efficacy of the solution procedure is demonstrated for a variety of problems in nonregular simply and multiply connected geometries with nonuniform boundary conditions.     

Nonlinear evolution of a first mode oblique wave in a compressible boundary layer: I. Heated/cooled walls

The nonlinear stability of an oblique mode propagating in atwo-dimensional compressible boundary layer is considered underthe long wavelength approximation. The growth rate of the waveis assumed to be small so that the ideas of unsteady nonlinearcritical layers can be applied. It is shown that the spatial/temporalevolution of the mode is governed by a pair of coupled unsteadynonlinear equations for the disturbance vorticity and density.Expressions for the linear growth rate show clearly the effectsof wall heating and cooling, and in particular how heating destabilizesthe boundary layer for these long wavelength inviscid modesat O(1) Mach numbers. A generalized expression for the lineargrowth rate is obtained and is shown to compare very well fora range of frequencies and wave angles at moderate Mach numberswith full numerical solutions of the linear stability problem.The numerical solution of the nonlinear unsteady critical layerproblem using a novel method based on Fourier decompositionand Chebyshev collocation is discussed and some results arepresented.     

Mathematical modelling and stabilization of large flexible spacecraft subject to orbital perturbation

In this paper the problem of modelling of large flexible spacecraft and their stabilization under the influence of orbital (radial) perturbation is considered. A complete dynamics of the spacecraft consisting of a rigid bus and a flexible beam is derived using Hamilton's principle. The equations of motion consist of a coupled system of partial differential equations governing the vibration of the flexible beam and ordinary differential equations describing the translational and rotational motions of the rigid bus. The asymptotic stability of the system is proved using Lyapunov's approach. Simple feedback controls are suggested for the stabilization of the system. For illustration, numerical simulations are carried out, giving interesting results.     

Implicit algorithms for solving the Cauchy problem for the equations of the dynamics of mechanical systems

It is shown that in the numerical solution of the Cauchy problem for systems of second-order ordinary differential equations, when solved for the highest-order derivative, it is possible to construct simple and economical implicit computational algorithms for step-by-step integration without using laborious iterative procedures based on processes of the Newton-Raphson iterative type. The initial problem must first be transformed to a new argument — the length of its integral curve. Such a transformation is carried out using an equation relating the initial parameter of the problem to the length of the integral curve. The linear acceleration method is used as an example to demonstrate the procedure of constructing an implicit algorithm using simple iterations for the numerical solution of the transformed Cauchy problem. Propositions concerning the computational properties of the iterative process are formulated and proved. Explicit estimates are given for an integration stepsize that guarantees the convergence of the simple iterations. The efficacy of the proposed procedure is demonstrated by the numerical solution of three problems. A comparative analysis is carried out of the numerical solutions obtained with and without parametrization of the initial problems in these three settings. As a qualitative test the problem of the celestial mechanics of the “Pleiades” is considered. The second example is devoted to modelling the non-linear dynamics of an elastic flexible rod fixed at one end as a cantilever and coiled in its initial (static) state into a ring by a bending moment. The third example demonstrates the numerical solution of the problem of the “unfolding” of a mechanical system consisting of three flexible rods with given control input.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Saturation-free numerical scheme for computing the flow past a lattice of airfoils and the determination of separation points in a viscous fluid

A numerical method for computing the potential flow past a lattice of airfoils is described. The problem is reduced to a linear integrodifferential equation on the lattice contour, which is then approximated by a linear system of equations with the help of specially derived quadrature formulas. The quadrature formulas exhibit exponential convergence in the number of points on an airfoil and have a simple analytical form. Due to its fast convergence and high accuracy, the method can be used to directly optimize the airfoils as based on any given integral characteristics. The shear stress distribution and the separation points are determined from the velocity distribution at the airfoil boundary calculated by solving the boundary layer equations. The method proposed is free of laborious grid generation procedures and does not involve difficulties associated with numerical viscosity at high Reynolds numbers.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

Numerical exploration of a system of reaction–diffusion equations with internal and transient layers

A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reaction–diffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method. The numerical results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem.     

Numerical analysis of a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.      

The reissner-sagoci problem for a multilayer base with a cylindrical cavity

We study the problem of the torsional oscillations of a plane disk-shaped die coupled with the upper boundary of a multilayer elastic base containing a vertical cylindrical cavity whose axis is perpendicular to the interface of the layers. The problem is stated as paired integral equations connected with the Weber integral transforms. To couple the solutions in the layers we use the method of initial parameters, which makes it possible to express the stress-strain state in any layer in terms of the solution of a Fredholm integral equation of second kind, to which the paired equations reduce. We exhibit an algorithm for numerical implementation of the problem. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 55–61.