a comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains abstract as an alternative to domain discretization methods the boundary element method bem provides a powerful tool for the calculation of dynamic structural response in frequency and time domain field equations of motion and boundary conditions are cast into boundary integral equations bie which are discretized only on the boundary fundamental solutions are used as weighting functions in the bie which fulfil the sommerfeld radiation condition i e the energy radiation into a surrounding medium is modelled correctly therefore infinite and semi infinite domains can be effectively treated by the method the soil represents such a semi infinite domain in soil structure interaction problems the response to vibratory loads superimposed to static pre loads can often be calculated by linear viscoelastic constitutive equations conventional viscoelastic constitutive equations can be generalized by taking fractional order time derivatives into account in the present paper two time domain bem approaches including generalized viscoelastic behaviour are compared with the laplace domain bem approach and subsequent numerical inverse transformation one of the presented time domain approaches uses an analytical integration of the elastodynamic bie in a time step viscoelastic constitutive properties are introduced after laplace transformation by means of an elastic viscoelastic correspondence principle the transient response is obtained by inverse transformation in each time step the other time domain approach is based on the so called convolution quadrature method in this formulation the convolution integral in the bie is numerically approximated by a quadrature formula whose weights are determined by the same laplace transformed fundamental solutions used in the first method and a linear multistep method a numerical study of wave propagation problems in 3 d viscoelastic continuum is performed for comparing the three bem formulations

numerical analysis of the lattice boltzmann method for simulation of linear acoustic waves we analyze a linear lattice boltzmann lb formulation for simulation of linear acoustic wave propagation in heterogeneous media we employ the single relaxation time bhatnagar gross krook as well as the general multirelaxation time collision operators by calculating the dispersion relation for various 2d lattices we show that the d2q5 lattice is the most suitable model for the linear acoustic problem we also implement a grid refinement algorithm for the lb scheme to simulate waves propagating in a heterogeneous medium with velocity contrasts our results show that the lb scheme performance is comparable to the classical second order finite difference schemes given its efficiency for parallel computation the lb method can be a cost effective tool for the simulation of linear acoustic waves in complex geometries and multiphase media

on the moments of the integrated geometric brownian motion abstract this note demonstrates how the divided differences characterization for the moments of the integrated geometric brownian process arises naturally from the solution to their differential equations the characterization was introduced by baxter and brummelhuis in their paper baxter and brummelhuis 2011 where they demonstrate its consistency with the results found by oshanin et al 1993 here we demonstrate its validity with more general expressions presented in other papers and explore the divided differences characterization of these moments further

accurate computation of divided differences of the exponential function abstract the traditional recurrence for the computation of exponential divided differences along with a new method based on the properties of the exponential function are studied in detail in this paper our results show that it is possible to combine these two methods to compute exponential divided differences accurately a hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference author

a massively parallel exponential integrator for advection diffusion models this work considers the real leja points method relpm m caliari m vianello l bergamaschi interpolating discrete advection diffusion propagators at spectral leja sequences j comput appl math 172 2004 79 99 for the exponential integration of large scale sparse systems of odes generated by finite element or finite difference discretizations of 3 d advection diffusion models we present an efficient parallel implementation of relpm for polynomial interpolation of the matrix exponential propagators exp dta v and f dta v f z exp z 1 z a scalability analysis of the most important computational kernel inside the code the parallel sparse matrix vector product has been performed as well as an experimental study of the communication overhead as a result of this study an optimized parallel sparse matrix vector product routine has been implemented the resulting code shows good scaling behavior even when using more than one thousand processors the numerical results presented on a number of very large test cases gives experimental evidence that relpm is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

an application of szego quadratures to the computation of the fourier transform abstract szego quadrature formulas represent the analogue on the unit circle of the well known gauss christoffel quadratures for intervals of the real line in the present work szego formulas are revised and an application to the computation of the fourier transform given

a steady state lattice boltzmann model for incompressible flows in this paper we propose a steady state lattice boltzmann equation which is independent of time for steady incompressible flows on the basis of the steady state lattice boltzmann equation we find a way of modeling some steady state problems using the lattice boltzmann model in further study we investigate steady state incompressible flows with the method of the higher order moment of equilibrium distribution functions and a series of partial differential equations for different space scales by assuming r constant we obtain the momentum equation for incompressible steady state flow the numerical results show that the new method is effective

fast and stable contour integration for high order divided differences via elliptic functions in this paper we will present a new method for evaluating high order divided differences for certain classes of analytic possibly operator values functions this is a classical problem in numerical mathematics but also arises in new applications such as e g the use of generalized convolution quadrature to solve retarded potential integral equations the functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

on the numerical stability of newton s formula for lagrange interpolation abstract it was previously noticed in the literature that the computation of newton s formula can be very sensitive to the ordering of the input data and it may be numerically unstable even for interpolation at the chebyshev nodes nevertheless to the best of our knowledge the effects of rounding errors in the computation of newton s formula have not been properly quantized yet in this note we make a full stability analysis of the propagation of rounding errors in the computation of newton s formula for lagrange interpolation we derive sharp upper bounds for the numerical error in the computations of three algorithms for computing polynomial interpolants in newton form that can be used to understand the instabilities mentioned above we throw light on and give a practical meaning to the condition number defined in carnicer et al 2019 by showing that it is the right thermometer for measuring the effects of rounding errors in the computation of newton s formula

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

on computing rational gauss chebyshev quadrature formulas we provide an algorithm to compute the nodes and weights for gauss chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside 1 1 contrary to existing rational quadrature formulas the computational effort is very low even for extremely high degrees and under certain conditions on the poles it can be shown that the complexity is of order o n this method is based on the derivation of explicit expressions for chebyshev orthogonal rational functions which are thus far the only examples of explicitly known orthogonal rational functions on 1 1 with arbitrary real poles outside this interval

runge kutta convolution quadrature methods for well posed equations with memory runge kutta based convolution quadrature methods for abstract well posed linear and homogeneous volterra equations non necessarily of sectorial type are developed a general representation of the numerical solution in terms of the continuous one is given the error and stability analysis is based on this representation which for the particular case of the backward euler method also shows that the numerical solution inherits some interesting qualitative properties such as positivity of the exact solution numerical illustrations are provided

on the quadrature error in operational quadrature methods for convolutions we analyze the quadrature error associated with operational quadrature methods for convolution equations the assumptions are that the convolution kernel is inl 1 and that its laplace transform is analytic and bounded in an obtuse sector of the complex plane under these circumstances the laplace transform has a slow variation property which admits a fourier analysis of the quadrature error we provide generall p error estimates assuming suitable smoothness conditions on the function under convolution

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

an algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes a multi time perturbation algorithm is derived to study certain synchronous control systems and nonlinear renewal processes the new methods presented here study singular perturbation problems for nonlinear volterra integro differential equations whose kernels are close to the dirac delta function the algorithm provides useful approximate solutions to weakly filtered phase locked loop circuits and to nonlinear renewal equations describing certain population dynamics

time domain boundary integral equation modeling of heat transmission problems this paper investigates the numerical modeling of a time dependent heat transmission problem by the convolution quadrature boundary element method it introduces the latest theoretical development into the error analysis of the numerical scheme semigroup theory is applied to obtain stability in the spatially semidiscrete scheme functional calculus is employed to yield convergence in the fully discrete scheme we compare the results to a more traditional laplace domain approach

matrix methods for quadrature formulas on the unit circle a survey in this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle like nodes and weights of gauss quadrature formulas for the estimation of integrals with respect to measures on the real line can be computed from the eigenvalue decomposition of the jacobi matrix szeg quadrature formulas for the approximation of integrals with respect to measures on the unit circle can be obtained from certain unitary five diagonal or unitary hessenberg matrices that characterize the recurrence for an orthogonal laurent polynomial basis these quadratures are exact in a maximal space of laurent polynomials orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles more general szeg quadrature formulas can be obtained that are exact in certain spaces of rational functions in this context the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator mobius transform of the same five diagonal or hessenberg matrices

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

on the fly backward error estimate for matrix exponential approximation by taylor algorithm abstract in this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on the fly without the need to precompute in high precision quantities related to specific accuracies in this way the scaling parameter s and the degree m of the truncated taylor series the underlying method are adapted to the input matrix and not to a class of matrices sharing some spectral properties such as with the classical backward error analysis the result is a very flexible method which can approximate the matrix exponential at any desired accuracy moreover the risk of overscaling that is the possibility to select a value s larger than necessary is mitigated by a new choice as the sum of two powers of two finally several numerical experiments in matlab with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives

canonical transforms and the efficient integration of quantum mechanical wave equations the integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry the wave function s energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations we demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude our approach includes the so called kramers henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency

computing a alpha log a and related matrix functions by contour integrals new methods are proposed for the numerical evaluation of f mathbf a or f mathbf a b where f mathbf a is a function such as mathbf a 1 2 or log mathbf a with singularities in infty 0 and mathbf a is a matrix with eigenvalues on or near 0 infty the methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving jacobi elliptic functions the convergence is geometric so that the computation of f mathbf a b is typically reduced to one or two dozen linear system solves which can be carried out in parallel

quadrature formulas associated with rational modifications of the chebyshev weight functions making use of the connection between quadrature formulas on the unit circle and the interval 1 1 recently established in 1 explicit expressions for the nodes and the weights of the gaussian formulas associated with rational modifications of the chebyshev weight functions are given some illustrative numerical examples are also presented

generalized convolution quadrature based on runge kutta methods in this paper we develop the runge kutta generalized convolution quadrature with variable time stepping for the numerical solution of convolution equations for time and space time problems and present the corresponding stability and convergence analysis for this purpose some new theoretical tools such as tensorial divided differences summation by parts with runge kutta differences and a calculus for runge kutta discretizations of generalized convolution operators such as an associativity property will be developed in this paper numerical examples will illustrate the stable and efficient behavior of the resulting discretization

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

solutions to the discrete airy equation application to parabolic equation calculations in the case of the equidistant discretization of the airy differential equation discrete airy equation the exact solution can be found explicitly this fact is used to derive a discrete transparent boundary condition tbc for a schrodinger type equation with linear varying potential which can be used in parabolic equation simulations in underwater acoustics and for radar propagation in the troposphere we propose different strategies for the discrete tbc and show an efficient implementation finally a stability proof for the resulting scheme is given a numerical example in the application to underwater acoustics shows the superiority of the new discrete tbc

uniform error estimates of galerkin methods for monotone abel volterra integral equations on the half line we consider galerkin methods for monotone abel volterra integral equations of the second kind on the half line the l2 theory follows from kolodner s theory of monotone hammerstein equations we derive the loo theory from the l2 theory by relating the l2and loo spectra of operators of the form x b ax to one another here denotes convolution and b e l1 and a e loo as an extra condition we need b t o t a 1 with a 0 we also prove the discrete analogue in particular we verify that the galerkin matrix satisfies the discrete conditions

numerical validation of the quantum lattice boltzmann scheme in two and three dimensions the multidimensional formulation of the quantum lattice boltzmann qlb scheme is validated against analytical solutions of the time dependent nonrelativistic schrodinger equation in two and three spatial dimensions as a result we demonstrate the viability of the quantum lattice boltzmann scheme for the numerical solution of the time dependent schrodinger equation in multiple spatial dimensions in addition the links between the qlb scheme and sequential splitting methods for partial differential equations are also clarified

runge kutta convolution quadrature for the boundary element method abstract time domain boundary element formulations can be established either directly in time domain or via laplace or fourier domain somewhere in between are the convolution quadrature based boundary element formulations which utilize the laplace domain fundamental solution but establish a time stepping procedure up to now in applications mostly backward differential formulas of second order are used as the underlying multistep method however in recent mathematical literature also runge kutta methods have been applied here the use of runge kutta methods is explained in detail and some numerical studies are given in these studies the backward difference based procedures are compared to runge kutta methods for a non smooth problem an l 2 norm of the error is used as the basis of comparison the convergence of which is investigated theoretically as well the results confirm that the usage of the new techniques is preferable with regard to less numerical oscillations in the solution and better representation of wave fronts

an analysis of the modified l1 scheme for time fractional partial differential equations with nonsmooth data we introduce a modified l1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous c

lattice boltzmann model for high order nonlinear partial differential equations in this paper a general lattice boltzmann lb model is proposed for the high order nonlinear partial differential equation with the form ensuremath partial t ensuremath phi ensuremath sum k 1 m ensuremath alpha k ensuremath partial x k mathrm ensuremath pi k ensuremath phi 0 1 ensuremath le k ensuremath le m ensuremath le 6 ensuremath alpha k are constant coefficients mathrm ensuremath pi k ensuremath phi are some known differential functions of ensuremath phi as some special cases of the high order nonlinear partial differential equation the classical m kdv equation kdv burgers equation mathrm k n n burgers equation kuramoto sivashinsky equation and kawahara equation can be solved by the present lb model compared to the available lb models the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved in addition we also conducted a detailed chapman enskog analysis and found that the high order nonlinear partial differential equation can be correctly recovered from the proposed lb model finally a large number of simulations are performed and it is found that the numerical results agree with the analytical solutions and usually the present model is also more accurate than the existing lb models h lai and c ma sci china ser g 52 1053 2009 h lai and c ma phys a amsterdam 388 1405 2009 for high order nonlinear partial differential equations

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

orthogonal polynomial expansions for the matrix exponential many different algorithms have been suggested for computing the matrix exponential in this paper we put forward the idea of expanding in either chebyshev legendre or laguerre orthogonal polynomials in order for these expansions to converge quickly we cluster the eigenvalues into diagonal blocks and accelerate using shifting and scaling

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation

numerical simulation for solitary wave of klein gordon zakharov equation based on the lattice boltzmann model abstract this work reports a study of klein gordon zakharov equation by using the lattice boltzmann method the klein gordon zakharov equation describes the interaction of the langmuir wave and the ion acoustic wave in plasma numerical simulations of the constructed lattice boltzmann model for the klein gordon zakharov equation show the propagation and the interaction of the 1 1 and 2 1 dimensional solitary waves these simulation results are examined and compared by using different parameters and different schemes this work indicates that the lattice boltzmann method can be a very effective tool for the simulation of klein gordon zakharov equation

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method