The paper deals with a numerical solution of a wave equation. The solution algorithm uses optimal parameters which are obtained by using Laguerre transform in time for the wave equation. Additional parameters are introduced into a difference scheme of 2nd-order approximation for the equation. The optimal values of these parameters are obtained by minimizing the error of a difference approximation of the Helmholtz equation. Applying the inverse Laguerre transform in the equation for harmonics, a differential-difference wave equation with the optimal parameters is obtained. This equation is difference in the spatial variables and differential in time. An iterative algorithm for solving the differential-difference wave equation with the optimal parameters is proposed. The results of numerical calculations of the differential-difference equations for 2-dimensional and 1-dimensional versions of the equation are presented. It is shown that the difference schemes with the optimal parameters give an increase in the accuracy of solving the equations.

The paper considers difference schemes with optimal parameters for solving Maxwell’s equations. Using Laguerre transforms, the numerical values of the optimal parameters are determined and differential-difference equations are constructed. Differential-difference equations are solved by the finite-difference method with iterations over small optimal parameters. Optimal second-order difference schemes for one-dimensional and two-dimensional Maxwell’s equations are considered. Optimal parameters of difference schemes are given. It is shown that the use of optimal difference schemes leads to an increase in the accuracy of solution.

In this paper, optimal difference schemes for solving Maxwell's equations using a Laquerre spectral transformation are considered. Additional parameters are introduced into the difference scheme of equations for the harmonics. Numerical values of these parameters are obtained by minimizing the error of a difference approximation of a Helmholtz equation. The thus obtained optimal parameter values are used in constructing optimal difference schemes. Two versions of optimal difference schemes are considered. It is shown that the optimal difference schemes increase the accuracy of solution of the equations. A simple modification of the difference schemes increases the efficiency of the algorithm.

Optimal difference schemes based on the Laguerre transform are proposed for solving the wave equation. Additional parameters are introduced into the difference scheme used for the equations of harmonics. Numerical values of these parameters are obtained by minimizing the error in the difference approximation of the Helmholtz equation. The optimal parameter values thus obtained are used to construct optimal difference schemes. Optimal difference schemes of second- and fourth-order accuracy are considered. The optimal parameters of the difference schemes are presented. Their values depend only on the ratio of the spatial step sizes. It is shown that the use of optimal difference schemes improves the accuracy of solutions. The efficiency of the algorithm is enhanced by applying a simple modification of the difference scheme.

This paper deals with a difference scheme of second-orderapproximation using Laquerre transform for the one-dimensional Maxwellequations. Supplementary parameters are introduced into this differencescheme. These parameters are obtained by minimizing the error of adifference approximation for a Helmholtz equation. The optimalparameters do not depend on the step size and the number of nodes in thedifference scheme. It is shown that the use of the Laguerredecomposition method allows obtaining higher accuracy of approximationof the equations in comparison with similar difference schemes whenusing the Fourier decomposition method. The second-order finitedifference scheme with the parameters is compared to a fourth-orderdifference scheme in two cases: The use of the optimal difference schemewhen solving a problem of electromagnetic pulse propagation in aninhomogeneous medium yields a solution accuracy comparable to thatobtained with the fourth-order difference scheme. When solving aninverse problem the second-order difference scheme allows obtaininghigher solution accuracy as compared to the fourth-order differencescheme. In these problems the second-order difference scheme with thesupplementary parameters has decreased the calculation time by 20–25% ascompared to the fourth-order difference scheme.

In this paper, a solution to a two-dimensional wave equation using the Laguerre transform is considered. Optimal parameters of finite difference schemes for this equation are obtained. Numerical values of these optimal parameters are specified. Second-order finite difference schemes with the optimal parameters provide an accuracy of solving the equations close to that provided by a fourth-order scheme. It is shown that using the Laguerre decomposition can reduce the number of optimal parameters in comparison with using the Fourier decomposition. This simplifies the finite difference schemes and decreases the number of calculations, that is, makes the algorithm more efficient.

A multilevel algorithm for solving the inverse problem for the diffusion equation by the optimization method using Laguerre functions is considered. Numerical calculations are carried out for Maxwell’s equations in a one-dimensional formulation in the diffusion approximations. Using the known solution, we find the medium’s conductivity distribution at some point of space. The function of the Laguerre harmonics is minimized by Newton’s method and the conjugate gradients technique. We investigate the influence of the form of the source of the electromagnetic waves and its spectrum on the solution’s accuracy for the inverse problem. The accuracy of this solution obtained by a multi-level algorithm and using conventional single-level algorithms is compared.

In this paper, the solution of two-dimensional Maxwell’s equations is considered using the Laguerre transform. Optimal parameters of the difference schemes for the equations are obtained and presented. Numerical values of these optimal parameters are given. Second-order difference schemes with the optimal parameters provide an accuracy of the solution of the equations that is comparable to the accuracy of the solution using fourth-order schemes. It is shown that, when using the Laguerre transform, the number of optimal parameters can be reduced compared to the Fourier transform. This reduction leads to a simplification of the difference scheme and a reduction in the amount of computation, i.e., to efficiency of the algorithm.

An inverse problem is solved by an optimization method using Laguerre functions. Numerical simulations are carried out for one-dimensional Maxwell’s equations in the wave and diffusion approximations. The spatial distributions of permittivity and conductivity in a medium are determined from a known solution at a certain point. A Laguerre harmonics function is minimized. The minimization is performed by the conjugate gradient method. The results of determining permittivity and conductivity are presented. The influence of the shape and spectrum of a source of electromagnetic waves on the solution accuracy of the inverse problem is investigated. The solutions with broadband and harmonic sources of electromagnetic waves are compared in accuracy.

A spectral method for modeling high-frequency electromagnetic waves in axisymmetric geometry is proposed. The method is based on the expansion of the solutions of Maxwell's equations in Laguerre functions in the time region.The spectral method is used to solve Maxwell's equations for both 2D media and stratified media. In the case of stratified media, a Fourier-Bessel expansion in the radial variable is used. The effectiveness of the spectral and finite-difference methods is compared. Harmonic solutions and solitary solutions by the Laguerre method are considered, and the dynamics of monochromatic and broadband electromagnetic pulses are examined. (C) 2011, V. S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B. V. All rights reserved.

A numerical solution to an inverse problem for the acoustic equations using an optimization method for a stratified medium is presented. With the distribution of an acoustic wave field on the medium’s surface, the 1D distributions of medium’s density, as well as the velocity and absorption coefficient of the acoustic wave, are determined. Absorption in a Voigt body model is considered. The conjugate gradients and the Newton method are used for minimization. To increase the efficiency of the numerical method, a multilevel adaptive algorithm is proposed. The algorithm is based on a division of the whole procedure of solving the inverse problem into a series of consecutive levels. Each level is characterized by the number of parameters to be determined at the level. In moving from one level to another, the number of parameters changes adaptively according to the functional minimized and the convergence rate. The minimization parameters are chosen as illustrated by results of solving the inverse problem in a spectral domain, where the desired quantities are presented as Chebyshev polynomial series and minimization is carried out with respect to the coefficients of these series. The method is compared in efficiency with a nonadaptive method. The optimal parameters of the multilevel method are chosen. It is shown that the multilevel algorithm offers several advantages over the one without partitioning into levels. The algorithm produces primarily a more accurate solution to the inverse problem.

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.

We apply the spectral Laguerre transform in the time domain to solve 2D Maxwell's equations for propagation of electromagnetic waves in lossy anisotropic media. The new algorithm is simple and efficient as Maxwell's equations are reduced to a harmonic series of linear algebraic equations where the matrix is independent of the harmonic order and is the same for all harmonics.The efficiency of the algorithm is improved by fitting a specially introduced free parameter of the Laguerre transform. If it is large, the solution spectrum shifts toward higher harmonics which is formally equivalent to the case of ray approximation.The Laguerre solution is comparable with high-order accurate finite-difference time-domain (FDTD) solutions. The method is stable both in the region of the wavefield, where conductivity approaches zero and the spectral Fourier method is unstable, and in the high-conductivity region, where the explicit FDTD code requires a too small time step. (C) 2008, IGM, Siberian Branch of the RAS. Published by Elsevier B. V. All rights reserved.

An inverse solution to the ID wave equation is obtained using the spectral Laguerre transform to find the distribution of wave velocities at some point of the medium. The problem is solved as optimization in which the function of Laguerre harmonics is minimized by the conjugate gradient or Newton's algorithms. Reported are velocities of a wave defined by a stepwise constant function. The accuracy of the inverse Solution for the Laguerre harmonics is investigated against the approximation accuracy in the boundary problem. The accuracy and efficiency of the Laguerre method are compared to those in the Fourier method. (c) 2007, IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.

We model propagation of electromagnetic waves in frequency-dependent media applying the Laguerre transform in time domain. The new algorithm is fourth-order accurate in space and computationally efficient. Maxwell's equations are reduced to a harmonic series of linear algebraic equations in which only the right side depends on the harmonic number and the inverse matrix is the same for all harmonics. The efficiency of computation for the algebraic equations is improved by fitting the free parameter of the Laguerre transform. The value of this parameter is easy to find and is likewise the same for all harmonics. The Laguerre scheme provides a better accuracy than the second-order accurate finite-difference solution at large path lengths. The method is stable both in the region of the wavefield where conductivity approaches zero and the spectral Fourier method is unstable, and in the high-conductivity region where the explicit FDTD code requires a too small time step.

One-dimensional Maxwell's equations with relaxation are solved using the Laguerre time series. Space derivatives are obtained by difference approximation of the fourth order of accuracy. The equations reduce to a harmonic series of linear algebraic equations, in which each harmonic is obtained by multiplication of the matrix by the right term. The matrix is universal and is the same for all harmonics. The algorithm shows a high efficiency.With the optimum choice of parameters, the Laguerre transform is advantageous over the finite-difference and Fourier methods. It is more precise than the second order approximation of the finite-difference solution, and uncertainty grows at a lower rate.The method is applicable to two- and three-dimensional Maxwell's equations, without significant limitations.

The inverse problem of the acoustic equation for attenuation in the frequency region is solved by the optimization method. A spectral multilevel adaptive algorithm with expansion of the unknown quantities into a Fourier series is proposed.Comparison is made between the efficiencies of spectral adaptive variants of three iteration minimization methods: gradient method, conjugate-gradient method, and Newton method. It is shown that transition to the spectral region increases the convergence of both adaptive and nonadaptive algorithms. This transition is most profitable for the gradient and conjugate-gradient methods.A spectral multilevel adaptive algorithm requires 2-5 times less calculations than a nonadaptive one. Compared with both nonspectral adaptive and nonadaptive algorithms, a spectral algorithm provides a more precise solution to the inverse problem for the gradient and conjugate-gradient methods.

An inverse problem for equations for acoustic-wave attenuation is solved by the optimization method. To increase the precision and efficiency of solution, an adaptive algorithm with the variable number of parameters is proposed. Three adaptive methods of minimization - gradient method, conjugate-gradient method, and Newton's method - are compared by efficiency. It is demonstrated that the adaptive algorithm significantly increases the convergence of these methods. The increase is irregular and most pronounced for a conjugate-gradient method. This method is more efficient than the other two. On average, the adaptive variant requires 2-4 times less calculations than the nonadaptive one. The adaptive algorithm increases the precision of environmental parameters for any of the above three minimization methods.

An inverse problem of electrical prospecting is considered for a harmonical source in case of two-dimensional conductivity using an optimization method. The quantity to be minimized is a squared deviation of the measured field from the calculated field obtained for some approximate distribution of conductivity al points of observation on the surface. The minimum point is sought by the method of joint gradients. To reduce the calculation time, an adaptive algorithm is proposed to solve the problem. The recovery efficiency of conductivity distribution in spate is studied as a Function of number and disposition of field observation points on the surface. Estimates are given to values of the functional to be minimized and precision of conductivity recovery. These relationships are illustrated by results of numerical calculations.