Abstract: Publication date: Available online 3 June 2016
Source:Applied Numerical Mathematics
Author(s): Leonardo Di G. Sigalotti, Jaime Klapp, Otto Rendón, Carlos A. Vargas, Franklin Peña-Polo
The problem of consistency of smoothed particle hydrodynamics (SPH) has demanded considerable attention in the past few years due to the ever increasing number of applications of the method in many areas of science and engineering. A loss of consistency leads to an inevitable loss of approximation accuracy. In this paper, we revisit the issue of SPH kernel and particle consistency and demonstrate that SPH has a limiting second-order convergence rate. Numerical experiments with suitably chosen test functions validate this conclusion. In particular, we find that when using the root mean square error as a model evaluation statistics, well-known corrective SPH schemes, which were thought to converge to second, or even higher order, are actually first-order accurate, or at best close to second order. We also find that observing the joint limit when N → ∞ , h → 0 , and n → ∞ , as was recently proposed by Zhu et al., where N is the total number of particles, h is the smoothing length, and n is the number of neighbor particles, standard SPH restores full C 0 particle consistency for both the estimates of the function and its derivatives and becomes insensitive to particle disorder.

Abstract: Publication date: Available online 7 June 2016
Source:Applied Numerical Mathematics
Author(s): Po-Hsien Lin, S.-T. John Yu
A set of model equations are proposed to simulate waves generated by unsteady, low-speed, nearly incompressible air and water flows. The equations include the continuity and momentum equations with pressure and velocity as the unknowns. Compressibility effect associated with waves motion is directly tracked by time-accurate calculation of pressure fluctuations. The corresponding density changes are modeled by using the bulk modulus of the medium. The three-dimensional equations are shown to be hyperbolic by analyzing eigenvalues and eigenvectors of the composite Jacobian matrix of the equations. Specifically, the matrix is shown to be diagonalizable and have a real spectrum. Moreover, an analytical form of the Riemann invariants of the one-dimensional equations are derived. To validate the model equations, the space-time Conservation Element and Solution Element (CESE) method and the SOLVCON code are employed to solve the two-dimensional equations. Aeolian tones generated by air and water flows passing a cylinder and over an open cavity are simulated. Numerical results compare well with previously reported data.

Abstract: Publication date: Available online 9 June 2016
Source:Applied Numerical Mathematics
Author(s): Sonia Seyed Allaei, Teresa Diogo, Magda Rebelo
We consider a general class of nonlinear singular Hammerstein Volterra integral equations. In general, these equations will have kernels containing both an end point and an Abel-type singularity, with exact solutions being typically nonsmooth. Under certain conditions, a uniformly convergent iterative solution is obtained on a small interval near the origin. In this work, two product integration methods are proposed and analysed where the integral over a small initial interval is calculated analytically, allowing the optimal convergence rates to be achieved. This is illustrated by some numerical examples.

Abstract: Publication date: Available online 11 June 2016
Source:Applied Numerical Mathematics
Author(s): Carlos J.S. Alves, Nuno F.M. Martins, Svilen S. Valtchev
Two meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.

Abstract: Publication date: Available online 9 June 2016
Source:Applied Numerical Mathematics
Author(s): Veronika Schleper
We present a new HLL-type approximate Riemann solver for a compressible two-phase flow model with phase transition and surface forces such as surface tension or electric forces. The solver is obtained following the main ideas of the HLL-approach. Due to the nonlinearity in the kinetic relation driving the phase transition, this solver involves the solution of one single nonlinear equation, contrary to a single-phase HLL-type solver, where no nonlinear equations have to be solved. We present some illustrative numerical examples to show the performance and accuracy of the new solver, comparing it to the relaxation Riemann solver from the literature as well as to the two-shock approximation, which is the most accurate interface Riemann solver currently available for the present situation, but suffers from high computational costs.

Abstract: Publication date: Available online 11 June 2016
Source:Applied Numerical Mathematics
Author(s): M. Kordy, E. Cherkaev, P. Wannamaker
This work develops a model order reduction method for a numerical solution of an inverse multi-frequency eddy current problem using a rational interpolation of the transfer function in the complex plane. We use the Pade interpolation in the complex frequency plane; this allows us to speed up the calculation of the frequency-dependent Jacobian in the inversion procedure without loosing accuracy. Interpolating frequencies are chosen adaptively to reduce the maximal approximation error. We use the error indicator that is equivalent to a seminorm of the residual. The efficiency of the developed approach is demonstrated by applying it to the inverse magnetotelluric problem, which is a geophysical electromagnetic remote sensing method used in mineral, geothermal, and groundwater exploration. In this application, the transfer function values are needed for shifts in a purely imaginary interval. Thus we consider the interpolating shifts in the same interval as well as in a purely real interval, containing the spectrum of the operator. Numerical tests show an excellent performance of the proposed methods characterized by a significant reduction of computational time without loss of accuracy of the calculated Jacobian.

Abstract: Publication date: September 2016
Source:Applied Numerical Mathematics, Volume 107
Author(s): S. Khodayari-Samghabadi, S.H. Momeni-Masuleh, A. Malek
In this paper, we present a stabilized explicit-extended penalty Galerkin method based on the implicit pressure and explicit saturation method to find the global solution for the two-phase flow in porous media at each time step. The bubble functions are employed as basis of the spatial dimensions for the extended penalty Galerkin method. The forward Euler method is applied to the temporal discretization. Since the accuracy of numerical simulations flow through porous media depends on the modeling of the injection and production well, we propose a new well model for the presented method. The details of the stability analysis for the proposed method are provided and suitable values of the penalty term and time steps are calculated. The efficiency of the method is illustrated by simulations of a waterflood in a heterogeneous oil reservoir. Comparisons are made with available literature which show the efficiency and accuracy of the proposed method.

Abstract: Publication date: Available online 11 June 2016
Source:Applied Numerical Mathematics
Author(s): C.F. Bracciali, A. Sri Ranga, A. Swaminathan
When a nontrivial measure μ on the unit circle satisfies the symmetry d μ ( e i ( 2 π − θ ) ) = − d μ ( e i θ ) then the associated orthogonal polynomials on the unit circle, say Φ n , are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials { z Φ n ( z ) + Φ n ⁎ ( z ) } and { z Φ n ( z ) − Φ n ⁎ ( z ) } , where Φ n ⁎ ( z ) = z n Φ n ( 1 / z ‾ ) ‾ , satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [ − 1 , 1 ] . The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials z Φ n ( z ) − Φ n ⁎ ( z ) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials z Φ n ( z ) + Φ n ⁎ ( z ) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫ 0 2 π e i θ − w − 2 d μ ( e i θ ) , where w is such that w = 1 , is given in terms of the coefficients α n − 1 = − Φ n ( 0 ) ‾ ... PubDate: 2016-06-16T18:07:53Z

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Sanjib Kumar Acharya, Ajit Patel
In this article, a class of second order parabolic initial-boundary value problems in the framework of primal hybrid principle is discussed. The interelement continuity requirement for standard finite element method has been alleviated by using primal hybrid method. Finite elements are constructed and used in spatial direction, and backward Euler scheme is used in temporal direction for solving fully discrete scheme. Optimal order estimates for both the semidiscrete and fully discrete method are derived with the help of modified projection operator. Numerical results are obtained in order to verify the theoretical analysis.

Abstract: Publication date: September 2016
Source:Applied Numerical Mathematics, Volume 107
Author(s): Tomoyuki Miyaji, Paweł Pilarczyk, Marcio Gameiro, Hiroshi Kokubu, Konstantin Mischaikow
We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself.

Abstract: Publication date: September 2016
Source:Applied Numerical Mathematics, Volume 107
Author(s): Jochen Schütz, Klaus Kaiser
In this publication, we consider IMEX methods applied to singularly perturbed ordinary differential equations. We introduce a new splitting into stiff and non-stiff parts that has a direct extension to systems of conservation laws and investigate its performance analytically and numerically. We show that this splitting can in some cases improve the order of convergence, demonstrating that the phenomenon of order reduction is not only a consequence of the method but also of the splitting.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Haibing Wang, Jijun Liu
Consider the scattering of long ocean tidal waves by an island taking into account the influence of daily rotation of the Earth, which is modeled by an exterior boundary value problem for the two-dimensional Helmholtz equation with generalized oblique derivative boundary condition. In this paper, we are concerned with a corresponding inverse scattering problem which is to reconstruct the unknown obstacle (island) from the far-field data. After proving the unique solvability of the direct scattering problem in a suitable function space required for our inverse scattering problem, we establish the linear sampling method (LSM) for reconstructing the boundary of the obstacle from the far-field data. To clarify the validity of such a sampling-type method which essentially depends on the solvability of an interior boundary value problem, we show that, except a discrete set of wave numbers, such an interior problem has a unique solution. Finally, some numerical examples are presented to demonstrate the efficiency of the reconstruction scheme.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Yifen Ke, Changfeng Ma
In this note, a technical error is pointed out in the proof of Theorem 3 in the paper M. Benzi and X.-P. Guo [1]. A correct proof of this theorem is given.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Igor Boglaev
The paper deals with numerical solution of coupled systems of nonlinear parabolic equations based on a nonlinear ADI scheme. The convergence of the nonlinear ADI scheme to the continuous solution is proved. A monotone iterative ADI method is constructed. The existence and uniqueness of a solution of the nonlinear ADI scheme are established. An analysis of convergence of the monotone iterative ADI method to the solution of the nonlinear ADI scheme on the whole time interval is given. Numerical experiments are presented.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): John T. Betts, Stephen L. Campbell, Karmethia C. Thompson
The numerical treatment of optimal control problems with state and control delays is important in a wide variety of scientific and technical applications. Solutions to these types of problems are difficult to obtain via analytic techniques since the system may be nonlinear and subjected to complicated inputs and constraints. There are several numerical methods available to compute the solutions of optimal control problems without delays. One such popular method is direct transcription. Although the numerical solutions of optimal control delay problems are important, less literature and software exists in this area. A general purpose industrial grade direct transcription code that can handle optimal control problems with both state and control constraints and delays is under development. Control delays pose a special challenge. A new technique for treating control delays when using a direct transcription approach is investigated in this paper.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Cong Shi, Chen Wang, Ting Wei
In this paper, we consider a class of severely ill-posed backward problems for linear parabolic equations. We use a convolution regularization method to obtain a stable approximate initial data from the noisy final data. The convergence rates are obtained under an a priori and an a posteriori regularization parameter choice rule in which the a posteriori parameter choice is a new generalized discrepancy principle based on a modified version of Morozov's discrepancy principle. The log-type convergence order under the a priori regularization parameter choice rule and log log -type order under the a posteriori regularization parameter choice rule are obtained. Two numerical examples are tested to support our theoretical results.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Raul Borsche
In this paper we propose a procedure to extend classical numerical schemes for hyperbolic conservation laws to networks of hyperbolic conservation laws. At the junctions of the network we solve the given coupling conditions and minimize the contributions of the outgoing numerical waves. This flexible procedure allows us to also use central schemes at the junctions. Several numerical examples are considered to investigate the performance of this new approach compared to the common Godunov solver and exact solutions.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Jing An
In this paper we present an efficient spectral method based on the Legendre–Galerkin approximation for the transmission eigenvalue problem. A rigorous error analysis is presented by using the minmax principle for the generalized eigenvalue problems associated to a transmission eigenvalue problem. However, this formulation can only compute real eigenvalues. Thus, we also present another formulation based on second order equations and construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. For the case of constant medium, we derive the matrix formulations based on the tensor-product for the discrete variational form in two and three-dimensional cases, respectively. In addition, we also establish an optimization scheme based on the Legendre–Galerkin approximation. With this scheme we can estimate the index of refraction of an inhomogeneous medium. We also present ample numerical results to show that our method is very effective and high accurate.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Changfeng Ma, Na Huang
By reformulating a class of weakly nonlinear complementarity problems as implicit fixed-point equations based on splitting of the system matrix, a modified modulus-based matrix splitting algorithm is presented. The convergence analysis of proposed algorithm is established for the case that the splitting of the system matrix is an H-splitting. Numerical experiments on two model problems are given to illustrate the theoretical results and examine the numerical effectiveness.

Abstract: Publication date: October 2016
Source:Applied Numerical Mathematics, Volume 108
Author(s): Peter Benner, Matthias Heinkenschloss, Jens Saak, Heiko K. Weichelt
This paper improves the inexact Kleinman–Newton method for solving algebraic Riccati equations by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman–Newton step. A convergence result is presented that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the Lyapunov equation residual, which in general do not hold for low-rank approaches. In the convergence proof of this paper, the line search is needed to ensure that the Riccati residuals decrease monotonically in norm. In the numerical experiments, the line search can lead to substantial reduction in the overall number of ADI iterations and, therefore, overall computational cost.

Abstract: Publication date: August 2016
Source:Applied Numerical Mathematics, Volume 106
Author(s): Fikriye Yılmaz, Aytekin Çıbık
In this work, we apply a variational multiscale stabilization (VMS) to the optimal control problems of Navier–Stokes equations. We first obtain the optimality conditions by using Lagrange approach. After stating the continuous optimality system, we formulate the discrete problem by a projection-based stabilized finite element scheme. We give the stability estimates for both state and adjoint state variables. Then, we present the a priori error analysis. The main issue of this paper is to show the efficiency of the variational multiscale stabilization for this optimal control problem. We verify our results by numerical examples.

Abstract: Publication date: Available online 16 May 2016
Source:Applied Numerical Mathematics
Author(s): John W. Pearson
In this manuscript we consider the development of fast iterative solvers for Stokes control problems, an important class of PDE-constrained optimization problems. In particular we wish to develop effective preconditioners for the matrix systems arising from finite element discretizations of time-dependent variants of such problems. To do this we consider a suitable rearrangement of the matrix systems, and exploit the saddle point structure of many of the relevant sub-matrices involved – we may then use this to construct representations of these sub-matrices based on good approximations of their ( 1 , 1 ) -block and Schur complement. We test our recommended iterative methods on a distributed control problem with Dirichlet boundary conditions, and on a time-periodic problem.

Abstract: Publication date: August 2016
Source:Applied Numerical Mathematics, Volume 106
Author(s): Peiqi Huang, Mingchao Cai, Feng Wang
In this paper, we propose a two-grid finite element method for solving the mixed Navier–Stokes/Darcy model with the Beavers–Joseph–Saffman interface condition. After solving a coupled nonlinear problem on a coarse grid, we sequentially solve decoupled and linearized subproblems on a fine grid and then correct the solution on the same grid. Compared with the existing work on the two-grid methods for the coupled model, our two-grid method allows a much higher order scaling between the coarse grid size H and the fine grid size h. Specifically, if a k-th order discretization is applied, by using h = H 2 k + 1 k for k = 1 , 2 and h = H k + 1 k − 1 for k ≥ 3 , the final step two-grid solution errors in the energy norm are still optimal. Numerical experiments are also given to confirm the theoretical analysis.

Abstract: Publication date: August 2016
Source:Applied Numerical Mathematics, Volume 106
Author(s): K. Jbilou, A. Messaoudi
In the present paper we introduce new block extrapolation methods as generalizations of the well known vector extrapolation methods. We give expressions of the obtained approximations via the Schur complement and also propose an efficient implementation of these methods. Applications to linearly generated sequences are given and extensions to nonlinear problems are also given. Applications of the proposed block extrapolation methods to some nonlinear matrix equations are considered and some numerical examples are given.

Abstract: Publication date: August 2016
Source:Applied Numerical Mathematics, Volume 106
Author(s): Mahboub Baccouch
We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O ( h p + 1 ) convergence rate in the L 2 -norm when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the DG solution is O ( h 2 p + 1 ) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O ( h p + 2 ) super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the ( p + 1 ) -degree right Radau polynomial and the less significant part converges at O ( h p + 2 ) rate in the L 2 -norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori error estimates and prove that they converge to the true errors in the L 2 -norm under mesh refinement. The order of convergence is proved to be p + 2 . Finally, we prove that the global effectivity index in the L 2 -norm converges to unity at O ( h ) rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented.

Abstract: Publication date: Available online 11 May 2016
Source:Applied Numerical Mathematics
Author(s): Joachim Rang
It is well-known that one-step methods have order reduction if they are applied on stiff ODEs such as the example of Prothero–Robinson. In this paper we analyse the local error of Runge–Kutta and Rosenbrock–Wanner methods. We derive new order conditions and define with them B P R -consistency. We show that for strongly A-stable methods B P R -consistency implies B P R -convergence. Finally we analyse methods from literature, derive new B P R -consistent methods and present numerical examples. The numerical and analytical results show the influence of different properties of the methods and of different order conditions on the numerical error and on the numerical convergence order.

Abstract: Publication date: Available online 9 May 2016
Source:Applied Numerical Mathematics
Author(s): Antoine Tambue, Jean Medard T. Ngnotchouye
We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and also provide the full weak convergence rate for non-self-adjoint linear operator with additive noise. Key part of the proof does not rely on Malliavin calculus. For non-self-adjoint operators, we analyse the optimal strong error for spatially semi discrete approximations for both multiplicative and additive noise with truncated and non-truncated noise. Depending on the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

Abstract: Publication date: Available online 10 May 2016
Source:Applied Numerical Mathematics
Author(s): YuFeng Shi, Yan Guo
In this paper, we apply a maximum-principle-satisfying finite volume compact weighted scheme to numerical modelling traffic flow problems on networks. Road networks can be numerically model as a graph, whose edges are a finite number of roads that join at junctions. The evolution on each road is described by a scalar hyperbolic conservation law, and traffic distribution matrices are used to formulate coupling conditions at the network junctions. In order to achieve maximum-principle of the traffic density on each road, the maximum-principle-satisfying polynomial rescaling limiter is adopted. Numerical results for road networks with rich solution structures are presented in this work and indicate that the finite volume compact weighted scheme produces essentially non-oscillatory, maximum principle preserving and high resolution solutions.

Abstract: Publication date: Available online 10 May 2016
Source:Applied Numerical Mathematics
Author(s): Sabine Le Borne, Lusine Shahmuradyan
In several production processes, the distribution of particles dispersed in an environmental phase may be mathematically described by the solution of population balance equations. We are concerned with the development of efficient numerical techniques for the aggregation process: It invokes an integral term that is usually numerically expensive to evaluate and often dominates the total simulation cost. We describe an approach on locally refined nested grids to evaluate both the source and the sink terms in almost linear complexity (instead of quadratic complexity resulting from a direct approach). The key is to switch from a nodal to a wavelet basis representation of the density function. We illustrate the numerical performance of this approach, both in comparison to a discretization of piecewise constant functions on a uniform grid as well as to the fixed pivot method on a geometric grid.

Abstract: Publication date: Available online 2 May 2016
Source:Applied Numerical Mathematics
Author(s): G. Kreiss, B. Krank, G. Efraimsson
A zone of increasingly stretched grid is a robust and easy-to-use way to avoid unwanted reflections at artificial boundaries in wave propagating simulations. In such a buffer zone there are two main damping mechanisms, dissipation and under-resolution that turns a traveling wave into an evanescent wave. We present analysis in one and two space dimensions showing that evanescent decay through under-resolution is a very efficient way to damp waves. The analysis is supported by numerical computations.

Abstract: Publication date: July 2016
Source:Applied Numerical Mathematics, Volume 105
Author(s): Hai Bi, Hao Li, Yidu Yang
This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.

Abstract: Publication date: July 2016
Source:Applied Numerical Mathematics, Volume 105
Author(s): Julia Leibinger, Michael Dumbser, Uwe Iben, Isabell Wayand
Flexible tubes are widely used in modern industrial hydraulic systems as connections between different components like valves, pumps and actuators. For the design and the analysis of the temporal behavior of a hydraulic system, one therefore needs an accurate mathematical model that describes the fluid flow in a compliant duct. Hence, in this paper we want to model the fluid–structure-interaction (FSI) problem given by the axially symmetric flow of a compressible barotropic fluid that flows through flexible tubes made of vulcanized rubber. The material of the tube can be described by using a visco-elastic rheology, which takes into account the strain relaxation of the material. The resulting mathematical model consists in a one-dimensional system of nonlinear hyperbolic partial differential equations (PDE) with non-conservative products and algebraic source terms. To solve this system numerically, we apply the DOT method, which is a generalized path-conservative Osher-type Riemann solver for conservative and non-conservative hyperbolic PDE recently proposed in [23] and [22]. We provide numerical evidence that the proposed DOT Riemann solver is well-balanced for the governing PDE system under consideration. The method is compared to available quasi-exact solutions of the Riemann problem in the case of an elastic wall described by the Laplace law. It is also compared to available experimental data and exact solutions obtained in the frequency domain for a linear visco-elastic wall behavior. In all cases under investigation the proposed path-conservative finite volume scheme based on the DOT Riemann solver is able to produce very accurate results.

Abstract: Publication date: Available online 21 April 2016
Source:Applied Numerical Mathematics
Author(s): Zhiping Mao, Sheng Chen, Jie Shen
We consider numerical approximation of the Riesz Fractional Differential Equations (FDEs), and construct a new set of generalized Jacobi functions, J n − α , − α ( x ) , which are tailored to the Riesz fractional PDEs. We develop optimal approximation results in non-uniformly weighted Sobolev spaces, and construct spectral Petrov–Galerkin algorithms to solve the Riesz FDEs with two kinds of boundary conditions (BCs): (i) homogeneous Dirichlet boundary conditions, and (ii) Integral BCs. We provide rigorous error analysis for our spectral Petrov–Galerkin methods, which show that the errors decay exponentially fast as long as the data (right-hand side function) is smooth, despite that fact that the solution has singularities at the endpoints. We also present some numerical results to validate our error analysis.

Abstract: Publication date: Available online 1 April 2016
Source:Applied Numerical Mathematics
Author(s): Norikazu Saito, Yoshiki Sugitani, Guanyu Zhou
We consider the stationary Stokes equations under a unilateral boundary condition of Signorini's type, which is one of artificial boundary conditions in flow problems. Well-posedness is discussed through its variational inequality formulation. We also consider the finite element approximation for a regularized penalty problem. The well-posedness, stability and error estimates of optimal order are established. The lack of a coupled Babuška and Brezzi's condition makes analysis difficult. We offer a new method of analysis. Particularly, our device to treat the pressure is novel and of some interest. Numerical examples are presented to validate our theoretical results.

Abstract: Publication date: Available online 30 March 2016
Source:Applied Numerical Mathematics
Author(s): Zhendong Gu, Xiaojing Guo, Daochun Sun
We propose series expansion method to solve VIEs (Volterra integral equations) with smooth given functions, including weakly singular VIEs possessing unsmooth solution. The key step in proposed method is to approximate given functions by their own Chebyshev Gauss-Lobatto interpolation polynomials. Lubich's results play an important role in the proposed method for weakly singular VIEs. Convergence analysis is provided for proposed method. Numerical experiments are carried out to confirm theoretical results.

Abstract: Publication date: Available online 30 March 2016
Source:Applied Numerical Mathematics
Author(s): I. Arun, Murugesan Venkatapathi
Sommerfeld integrals relate a spherical wave from a point source to a convolution set of plane and cylindrical waves. This relation does not have analytical solutions but it submits to a solution by numerical integration. Among others, it is significant for theoretical studies of many optical and radiation phenomena involving surfaces. This approach is preferred over discretized computational models of the surface because of the many orders of increased computations involved in the latter. One of the most widely used and accurate methods to compute these solutions is the numerical integration of the Sommerfeld integrand over a complex contour. We have analyzed the numerical advantages offered by this method, and have justified the optimality of the preferred contour of integration and the choice of two eigenfunctions used. In addition to this, we have also analyzed four other approximate methods to compute the Sommerfeld integral and have identified their regions of validity, and numerical advantages, if any. These include the high relative permittivity approximation, the short distance approximation, the exact image theory and Fourier expansion of the reflection coefficient. We also finally compare these five methods in terms of their computational cost.

Abstract: Publication date: Available online 25 March 2016
Source:Applied Numerical Mathematics
Author(s): G. Landi, E. Loli Piccolomini, I. Tomba
We present a discrepancy-like stopping criterium for iterative regularization methods for the solution of linear discrete ill-posed problems. The presented criterium terminates the iterations of the iterative method when the residual norm of the computed solution becomes less or equal to the residual norm of a regularized Truncated Singular Value Decomposition (TSVD) solution. We present two algorithms for the automatic computation of the TSVD residual norm using the Discrete Picard Condition. The first algorithm uses the SVD coefficients while the second one uses the Fourier coefficients. In this work, we mainly focus on the Conjugate Gradient Least Squares method, but the proposed criterium can be used for terminating the iterations of any iterative regularization method. Many numerical tests on some selected one dimensional and image deblurring problems are presented and the results are compared with those obtained by state-of-the-art parameter selection rules. The numerical results show the efficiency and robustness of the proposed criterium.

Abstract: Publication date: Available online 21 March 2016
Source:Applied Numerical Mathematics
Author(s): M. Hubenthal, D. Onofrei
In previous works we considered the Helmholtz equation with fixed frequency k outside a discrete set of resonant frequencies, where it is implied that, given a source region D a ⊂ R d ( d = 2 , 3 ‾ ) and u 0 , a solution of the homogeneous scalar Helmholtz equation in a set containing the control region D c ⊂ R d , there exists an infinite class of boundary data on ∂ D a so that the radiating solution to the corresponding exterior scalar Helmholtz problem in R d ∖ D a will closely approximate u 0 in D c . Moreover, it will have vanishingly small values beyond a certain large enough “far-field” radius R. In this paper we study the minimal energy solution of the above problem (e.g. the solution obtained by using Tikhonov regularization with the Morozov discrepancy principle) and perform a detailed sensitivity analysis. In this regard we discuss the stability of the minimal energy solution with respect to measurement errors as well as the feasibility of the active scheme (power budget and accuracy) depending on: the mutual distances between the antenna, control region and far field radius R; value of the regularization parameter; frequency; location of the source.

Abstract: Publication date: Available online 22 March 2016
Source:Applied Numerical Mathematics
Author(s): Mansur I. Ismailov, Ibrahim Tekin
In this paper, the direct and inverse initial boundary value problems for a first order system of two hyperbolic equations are considered. The method of characteristics and the finite difference method are applied to the theoretical and numerical solutions of the direct problem, respectively. Moreover the suitability of the method of characteristics for the inverse problem of finding solely space-dependent coefficients and the finite difference method for solely time-dependent coefficients of the first order hyperbolic system are shown. The stability of the numerical method is supported by the examples.

Abstract: Publication date: Available online 21 March 2016
Source:Applied Numerical Mathematics
Author(s): Alexandra Koulouri, Ville Rimpiläinen, Mike Brookes, Jari P. Kaipio
In the inverse source problem of the Poisson equation, measurements on the domain boundaries are used to reconstruct sources inside the domain. The problem is an ill-posed inverse problem and it is sensitive to modelling errors of the domain. These errors can be boundary, structure and material property errors, for example. In this paper, we investigate whether the recently proposed Bayesian approximation error (BAE) approach could be used to alleviate the source estimation errors when an approximate model for the domain is employed. The BAE is based on postulating a probabilistic model for the uncertainties, in this case the geometry and structure of the domain, and to carry out approximate marginalization over these nuisance parameters. We particularly consider electroencephalography (EEG) source imaging as an application. EEG is a diagnostic brain imaging modality, and it can be used to reconstruct neural sources in the brain from electric potential measurements along the scalp. In the feasibility study, we assess to which degree one can recover from the modelling errors that are induced by the use of the three concentric circle head model instead of an anatomically accurate head model. The studied domain modelling errors include errors in the geometry of the exterior boundary and the structure of the interior. We show that, in particular with superficial dipole sources, the BAE yields estimates that can in some cases be considered adequately accurate. This would avoid the need for the extraction of the accurate head features which is conventionally carried out via expensive and time consuming auxiliary imaging modalities such as magnetic resonance imaging.

Abstract: Publication date: Available online 24 March 2016
Source:Applied Numerical Mathematics
Author(s): V.S. Sizikov, D.N. Sidorov
We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytic computation of the singular integral. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it make sense to apply regularization in case of highly noisy (about 10%) sources only. That is due to the fact that singular integral equations enjoy selfregularization property.

Abstract: Publication date: Available online 22 March 2016
Source:Applied Numerical Mathematics
Author(s): M.A.V. Pinto, C. Rodrigo, F.J. Gaspar, C.W. Oosterlee
In this work, incomplete factorization techniques are used as smoothers within a geometric multigrid algorithm on triangular grids. A local Fourier analysis is proposed to study the smoothing properties of these methods, as well as the asymptotic convergence of the whole multigrid procedure. With this purpose, two- and three-grid local Fourier analysis are performed. Several two-dimensional diffusion problems, including different kinds of anisotropy are considered to demonstrate the robustness of this type of methods.

Abstract: Publication date: Available online 2 March 2016
Source:Applied Numerical Mathematics
Author(s): Haihua Qin, Xiaodong Liu
We are concerned with the reconstruction of both the penetrable inhomogeneous medium and the buried impenetrable obstacle. Firstly, the classical linear sampling method is used to recover the support of the inhomogeneous medium, and then a modification of the linear sampling method is proposed for objects buried in a known layered medium. The main feature of our method is that it avoids using knowledge of the Green's function for the background media. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of our method.

Abstract: Publication date: Available online 3 March 2016
Source:Applied Numerical Mathematics
Author(s): C. Rodrigo, F.J. Gaspar, F.J. Lisbona
A general local Fourier analysis for overlapping block smoothers on triangular grids is presented. This analysis is explained in a general form for its application to problems with different discretizations. This tool is demonstrated for two different problems: a stabilized linear finite element discretization of Stokes equations and an edge-based discretization of the curl-curl operator by lowest-order Nédélec finite element method. In this latter, special Fourier modes have to be considered in order to perform the analysis. Numerical results comparing two- and three-grid convergence factors predicted by the local Fourier analysis to real asymptotic convergence factors are presented to confirm the predictions of the analysis and show their usefulness.

Abstract: Publication date: July 2016
Source:Applied Numerical Mathematics, Volume 105
Author(s): Chang-tao Sheng, Zhong-qing Wang, Ben-yu Guo
In this paper, we propose a multistep Legendre–Gauss spectral collocation method for the nonlinear Volterra functional integro-differential equations (VFIDEs) with vanishing delays. This method is easy to implement and possesses the high-order accuracy. We also provide a rigorous convergence analysis of the hp-version of the multistep spectral collocation method under H 1 -norm. Numerical results confirm the theoretical analysis.

Abstract: Publication date: July 2016
Source:Applied Numerical Mathematics, Volume 105
Author(s): Franziska Nestler
We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.