Abstract: Publication date: Available online 28 April 2014
Source:Applied Numerical Mathematics
Author(s): E.N. Aristova , B.V. Rogov
A fourth-order accurate (in space) bicompact scheme is proposed for solving the inhomogeneous stationary transport equation in two dimensions. The scheme is based on a minimal stencil consisting of two nodes in each dimension and is obtained as a stationary limit of bicompact schemes produced by the method of lines for the nonstationary transport equation. The set of unknowns for each two-dimensional cell consists of the node values of the solution function and its integrals over cell edges and the entire cell. A closed system of linear equations is obtained for all desired variables in each cell. This system is solved using the running calculation method, which reveals the characteristic properties of the transport equation without explicitly using characteristics. The numerical results are compared with the solution produced by a conservative-characteristic method applied to a similar set of variables. The advantages of the bicompact schemes are demonstrated.

Abstract: Publication date: Available online 9 May 2014
Source:Applied Numerical Mathematics
Author(s): Christian Rohde , Christoph Zeiler
The dynamics of two-phase flows depend crucially on interfacial effects like surface tension and phase transition. A numerical method for compressible inviscid flows is proposed that accounts in particular for these two effects. The approach relies on the solution of Riemann-like problems across the interface that separates the liquid and the vapour phase. Since the analytical solutions of the Riemann problems are only known in particular cases an approximative Riemann solver for arbitrary settings is constructed. The approximative solutions rely on the relaxation technique. The local well-posedness of the approximative solver is proven. Finally we present numerical experiments for radially symmetric configurations that underline the reliability and efficiency of the numerical scheme.

Abstract: Publication date: Available online 14 May 2014
Source:Applied Numerical Mathematics
Author(s): A. Zlotnik , A. Romanova
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov–Crank–Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L 2 -stability (in particular, L 2 -conservativeness) together with the error estimate O ( τ 2 + h 4 ) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip and solving of tridiagonal systems in its main direction is developed to implement the splitting method for general potential. We also engage the Richardson extrapolation in time to increase the error order with respect to time step and get the method of higher order both in space and time. Numerical results on the tunnel effect for smooth and discontinuous rectangular barriers are included together with the careful practical error analysis on refining meshes.

Abstract: Publication date: Available online 14 May 2014
Source:Applied Numerical Mathematics
Author(s): A.A. Abramov , L.F. Yukhno
A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.

Abstract: Publication date: Available online 1 June 2014
Source:Applied Numerical Mathematics
Author(s): A. Gulin
The finite differences schemes with weights for the heat conduction equation with nonlocal boundary conditions u ( 0 , t ) = 0 , γ ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( 1 , t ) are discussed, where γ is a given real parameter. On some interval γ ∈ ( γ 1 , γ 2 ) the spectrum of the differential operator contains three eigenvalues in the left complex half-plane, while the remaining eigenvalues are located in the right half-plane. Earlier only the case of one eigenvalue λ 0 located in the left half-plane was considered. The stability criteria of finite differences schemes is formulated in the subspace induced by stable harmonics.

Abstract: Publication date: Available online 2 June 2014
Source:Applied Numerical Mathematics
Author(s): T.A. Averina , A.L. Bondareva , G.I. Zmievskaya
Model phase transition /PT/ involves the formation of defects(voids or blisters), their migration into thin layers sample from SiC and Mo, accumulation of defects, and consequently, a change in the crystal lattice strain which leads to amorphization of materials. Mathematical model is related with solution of stochastic differential equations /SDEs/. The scheme used is a two-level modification of the asymptotically unbiased numerical method for solving SDEs in the sense of Stratonovich, which has second order mean-square convergence for SDEs with a single noise or for SDEs with additive noise. On example of computer simulation of porosity and amorphization lattice are to be discussed characteristics of phase transition at initial stage as well their influence on protective qualities of SiC thin layer cover subjected by radiation of X e + + ions.

Abstract: Publication date: Available online 6 June 2014
Source:Applied Numerical Mathematics
Author(s): Tomás P. Barrios , Edwin M. Behrens , María González
We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.

Abstract: Publication date: Available online 14 June 2014
Source:Applied Numerical Mathematics
Author(s): Lizhen Chen , Jie Shen , Chuanju Xu , Li-Shi Luo
An efficient parallel algorithm for the time dependent incompressible Navier-Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-compliment approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers.

Abstract: Publication date: September 2014
Source:Applied Numerical Mathematics, Volume 83
Author(s): Wen Li , Seak-Weng Vong , Xiao-fei Peng
In this paper, we give some structured perturbation bounds for generalized saddle point matrices and Hermitian block tridiagonal matrices. Our bounds improve some existing ones. In particular, the proposed bounds reveal the sensitivity of the eigenvalues with respect to perturbations of different blocks. Numerical examples confirm the theoretical results.

Abstract: Publication date: September 2014
Source:Applied Numerical Mathematics, Volume 83
Author(s): S. Hussain , F. Schieweck , S. Turek
In this paper, we discuss solution techniques of Newton-multigrid type for the resulting nonlinear saddle-point block-systems if higher order continuous Galerkin–Petrov ( cGP ( k ) ) and discontinuous Galerkin (dG(k)) time discretizations are applied to the nonstationary incompressible Navier–Stokes equations. In particular for the cGP ( 2 ) method with quadratic ansatz functions in time, which lead to 3rd order accuracy in the L 2 -norm and even to 4th order superconvergence in the endpoints of the time intervals, together with the finite element pair Q 2 / P 1 disc for the spatial approximation of velocity and pressure leading to a globally 3rd order scheme, we explain the algorithmic details as well as implementation aspects. All presented solvers are analyzed with respect to their numerical costs for two prototypical flow configurations.

Abstract: Publication date: October 2014
Source:Applied Numerical Mathematics, Volume 84
Author(s): Santtu Salmi , Jari Toivanen
We propose families of IMEX time discretization schemes for the partial integro-differential equation derived for the pricing of options under a jump–diffusion process. The schemes include the families of IMEX-midpoint, IMEX-CNAB and IMEX-BDF2 schemes. Each family is defined by a convex combination parameter c ∈ [ 0 , 1 ] , which divides the zeroth-order term due to the jumps between the implicit and explicit parts in the time discretization. These IMEX schemes lead to tridiagonal systems, which can be solved extremely efficiently. The schemes are studied through Fourier stability analysis and numerical experiments. It is found that, under suitable assumptions and time step restrictions, the IMEX-midpoint family is conditionally stable only for c = 0 , while the IMEX-CNAB and the IMEX-BDF2 families are conditionally stable for all c ∈ [ 0 , 1 ] . The IMEX-CNAB c = 0 scheme produced the smallest error in our numerical experiments.

Abstract: Publication date: October 2014
Source:Applied Numerical Mathematics, Volume 84
Author(s): M. Seydaoğlu , S. Blanes
We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time-dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods on several numerical examples and present some new improved schemes.

Abstract: Publication date: October 2014
Source:Applied Numerical Mathematics, Volume 84
Author(s): Mahboub Baccouch
In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a ( p + 1 ) -degree Radau polynomial, when p-degree piecewise polynomials with p ≥ 1 are used. This result allows us to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L 2 -norm under mesh refinement. The order of convergence is proved to be p + 5 / 4 . Finally, we prove that the global effectivity indices in the L 2 -norm converge to unity at O ( h 1 / 2 ) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

Abstract: Publication date: Available online 18 April 2014
Source:Applied Numerical Mathematics
Author(s): Wenxing Zhang , Deren Han , Suoliang Jiang
In this paper, we propose a novel alternating projection based prediction-correction method for solving the monotone variational inequalities with separable structures. At each iteration, we adopt the weak requirements for the step sizes to derive the predictors, which affords fewer trial and error steps to accomplish the prediction phase. Moreover, we design a new descent direction for the merit function in correction phase. Under some mild assumptions, we prove the global convergence of the modified method. Some preliminary computational results are reported to demonstrate the promising and attractive performance of the modified method compared to some state-of-the-art prediction-contraction methods.

Abstract: Publication date: Available online 19 April 2014
Source:Applied Numerical Mathematics
Author(s): Sergey Kabanikhin , Alemdar Hasanov , Igor Marinin , Olga Krivorotko , David Khidasheli
Tsunamis are gravitational, i.e. gravity-controlled waves generated by a given motion of the bottom. There are different natural phenomena, such as submarine slumps, slides, volcanic explosions, earthquakes, etc. that can lead to a tsunami. This paper deals with the case where the tsunami source is an earthquake. The mathematical model studied here is based on shallow water theory, which is used extensively in tsunami modeling. The inverse problem consists of determining an unknown initial tsunami source q ( x , y ) by using measurements f m ( t ) of the height of a passing tsunami wave at a finite number of given points ( x m , y m ) , m = 1 , 2 , … , M , of the coastal area. The proposed approach is based on the weak solution theory for hyperbolic PDEs and adjoint problem method for minimization of the corresponding cost functional. The adjoint problem is defined to obtain an explicit gradient formula for the cost functional J ( q ) = ‖ A q − F ‖ 2 , F = ( f 1 , … , f M ) . Numerical algorithms are proposed for the direct as well as adjoint problems. Conjugate gradient algorithm based on explicit gradient formula is used for numerical solution of the inverse problem. Results of computational experiments presented for the synthetic noise free and random noisy data in real scale illustrate bounds of applicability of the proposed approach, also its efficiency and accuracy.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Aurore Back , Eric Sonnendrücker
The notion of B-spline based discrete differential forms is recalled and along with a Finite Element Hodge operator, it is used to design new numerical methods for solving the Vlasov–Poisson system.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): V. Selgas
This paper deals with the numerical solution of the time-harmonic eddy current model in an axisymmetric unbounded domain. To this end, a new symmetric BEM–FEM formulation is derived and also analyzed. Moreover, error estimates for the corresponding discretization are proven.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Francesca Rapetti , Germain Rousseaux
In this paper we introduce the electromagnetic quasi-static models in a simple but meaningful way, relying on the dimensional analysis of Maxwell's equations. This analysis puts in evidence the three characteristic times of an electromagnetic phenomenon. It allows to define the range of validity of well-known models, such as the eddy-current (MQS) or the electroquasistatic (EQS) ones, and thus their pertinence to describe a given phenomenon. The role of the so-called “small parameters” of a model is explained in detail for two classical examples, namely a capacitor and a solenoid. We show how the MQS and EQS models result from having replaced fields by their first order truncations of Taylor expansions with respect to these small parameters. We finally investigate the connection between quasi-static models and circuit theory, clarifying the role of the fields with respect to classical circuit elements, and provide an example of application to study the electromagnetic fields in a simple case.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Ralf Hiptmair , Andrea Moiola , Ilaria Perugia
We extend the a priori error analysis of Trefftz discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non-convex domains with non-connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L 2 -error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Sébastien Imperiale , Patrick Joly
In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and electric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain “homogenized” telegraphist's model. This model, which includes a nonlocal in time operator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross section is not homogeneous.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Qiang Chen , Peter Monk
Time domain integral equations complement other methods for solving Maxwell's equations by handling infinite domains without difficulty and by reducing the computational domain to the surface of the scatterer. In this paper we study the discretization error when convolution quadrature is used to discretize two new regularized combined field integral equation formulations of the problem of computing scattering from a bounded perfectly conducting obstacle.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): A.-S. Bonnet-Ben Dhia , L. Chesnel , P. Ciarlet Jr.
We consider the theoretical study of time harmonic Maxwell's equations in presence of sign-changing coefficients, in a two-dimensional configuration. Classically, the problems for both the Transverse Magnetic and the Transverse Electric polarizations reduce to an equivalent scalar Helmholtz type equation. For this scalar equation, we have already studied consequences of the presence of sign-changing coefficients in previous papers, and we summarize here the main results. Then we focus on the alternative approach which relies on the two-dimensional vectorial formulations of the TM or TE problems, and we exhibit some unexpected effects of the sign-change of the coefficients. In the process, we provide new results on the scalar equations.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Daniele Boffi , Lucia Gastaldi
In this paper we investigate the behavior of the finite element approximation of multiple eigenvalues in presence of eigenfunctions with different smoothness. We start from a one-dimensional example presented in the Handbook of Numerical Analysis by Babuška and Osborn and extend it to higher order approximation and to two dimensions, confirming that the different regularities of the eigenfunctions are well seen in the numerical computations. Then we discuss a mixed formulation corresponding to the one-dimensional example. It turns out that the regularity properties of the eigenfunctions are not well separated in this particular example, since the estimates have to take into account both components of the solution.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Oszkár Bíró , Gergely Koczka , Kurt Preis
An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three-dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady-state periodic solution is of interest only. This is represented either in the frequency domain as a finite Fourier series or in the time domain as a set of discrete time values within one period for each finite element degree of freedom. The former approach is the (continuous) harmonic balance method and, in the latter one, discrete Fourier transformation will be shown to lead to a discrete harmonic balance method. Due to the nonlinearity, all harmonics, both continuous and discrete, are coupled to each other. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a time-independent permeability distribution, the so-called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps. As industrial applications, analyses of large power transformers are presented. The first example is the computation of the electromagnetic field of a single-phase transformer in the time domain with the results compared to those obtained by traditional time-stepping techniques. In the second application, an advanced model of the same transformer is analyzed in the frequency domain by the harmonic balance method with the effect of the presence of higher harmonics on the losses investigated. Finally a third example tackles the case of direct current (DC) bias in the coils of a single-phase transformer.

Abstract: Publication date: June 2014
Source:Applied Numerical Mathematics, Volume 80
Author(s): Jiansong Zhang , Danping Yang , Shuqian Shen , Jiang Zhu
Combining the modified method of characteristics with adjusted advection with a splitting positive definite mixed element scheme, we establish a new mixed finite element procedure for solving compressible miscible displacement in porous media. This procedure can preserve the mass conservation globally, the coefficient matrix of the mixed system is symmetric positive definite and the flux equation is separated from the pressure equation. We analyse the convergence and give an optimal L 2 -norm error estimate. Finally we present some numerical results to confirm our theoretical analysis.

Abstract: Publication date: June 2014
Source:Applied Numerical Mathematics, Volume 80
Author(s): Adimurthi , K Sudarshan Kumar , G.D. Veerappa Gowda
Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich,… satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.

Abstract: Publication date: June 2014
Source:Applied Numerical Mathematics, Volume 80
Author(s): Li Cai , Wenxian Xie , Yufeng Nie , Jianhu Feng
We introduce a high resolution fifth-order semi-discrete Hermite central-upwind scheme for multidimensional Hamilton–Jacobi equations. The numerical fluxes of the scheme are constructed by Hermite polynomials which can be obtained by using the short-time assignment of the first derivatives. The extensions of the proposed semi-discrete Hermite central-upwind scheme to multidimensional cases are straightforward. The accuracy, efficiency and stability properties of our schemes are finally demonstrated via a variety of numerical examples.

Abstract: Publication date: June 2014
Source:Applied Numerical Mathematics, Volume 80
Author(s): Peder Aursand , Steinar Evje , Tore Flåtten , Knut Erik Teigen Giljarhus , Svend Tollak Munkejord
We present first- and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation. We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge–Kutta and exponential methods. Through operator splitting, an application to granular–gas flow is provided.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Kirk M. Soodhalter , Daniel B. Szyld , Fei Xue
We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Youngmok Jeon , Do Young Kwak
An immersed nonconforming finite element method based on the flux continuity on intercell boundaries is introduced. The direct application of flux continuity across the support of basis functions yields a nonsymmetric stiffness system for interface elements. To overcome non-symmetry of the stiffness system we introduce a modification based on the Riesz representation and a local postprocessing to recover local fluxes. This approach yields a P 1 immersed nonconforming finite element method with a slightly different source term from the standard nonconforming finite element method. The recovered numerical flux conserves total flux in arbitrary sub-domain. An optimal rate of convergence in the energy norm is obtained and numerical examples are provided to confirm our analysis.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Pouria Assari , Hojatollah Adibi , Mehdi Dehghan
This article investigates a numerical scheme based on the radial basis functions (RBFs) for solving weakly singular Fredholm integral equations by combining the product integration and collocation methods. A set of scattered points over the domain of integration is utilized to approximate the unknown function by using the RBFs. Since the proposed scheme does not require any background mesh for its approximations and numerical integrations unlike other product integration methods, it is called the meshless product integration (MPI) method. The method can be easily implemented and its algorithm is simple and effective to solve weakly singular integral equations. This approach reduces the solution of linear weakly singular integral equations to the solution of linear systems of algebraic equations. The error analysis of the proposed method is provided. The validity and efficiency of the new technique are demonstrated through several tests.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Seungil Kim
In this paper, we consider a Cartesian PML approximation to resonance values of time-harmonic problems posed on unbounded domains in R 2 . A PML is a fictitious layer designed to find solutions arising from wave propagation and scattering problems supplemented with an outgoing radiation condition at infinity. Solutions obtained by a PML coincide with original solutions near wave sources or scatterers while they decay exponentially as they propagate into the layer. Due to rapid decay of solutions, it is natural to truncate unbounded domains to finite regions of computational interest. In this analysis, we introduce a PML in Cartesian geometry to transform a resonance problem (characterized as an eigenvalue problem with improper eigenfunctions) on an unbounded domain to a standard eigenvalue problem on a finite computational region. Truncating unbounded domains gives rise to perturbation of resonance values, however we show that eigenvalues obtained by the truncated problem converge to resonance values as the size of computational domain increases. In addition, our analysis shows that this technique is free of spurious resonance values provided truncated domains are sufficiently large. Finally, we present the results of numerical experiments with simple model problems.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): X. Liu , G.L. Zhang , M.Z. Liu
This paper deals with exponential stability of both analytic and numerical solutions to nonlinear impulsive differential equations. Instead of Lyapunov functions a new technique is used in the analysis. A sufficient condition is given under which the analytic solution is exponential asymptotically stable. The numerical solutions are calculated by Runge–Kutta methods and the corresponding stability properties are studied. It is proved that algebraically stable Runge–Kutta methods satisfying 1 − b T A − 1 e < 1 can preserve the stability of the equation. Finally some numerical experiments are given to illustrate the conclusion.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Avram Sidi
Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the singular integrals I ( 1 ) [ u ] = ∫ a b ( cot π ( x − t ) T ) u ( x ) d x and I ( 2 ) [ u ] = ∫ a b ( csc 2 π ( x − t ) T ) u ( x ) d x , with b − a = T and u ( x ) a T-periodic continuous function on R . These integrals are not defined in the regular sense, but are defined in the sense of Cauchy Principal Value and Hadamard Finite Part, respectively. With h = ( b − a ) / n , n = 1 , 2 , … , the numerical quadrature formulas Q n ( 1 ) [ u ] for I ( 1 ) [ u ] and Q n ( 2 ) [ u ] for I ( 2 ) [ u ] are Q n ( 1 ) [ u ] = h ∑ j = 1 n f ( t + j h − h / 2 ) , f ( x ) = ( cot π ( x − t ) T ) u ( x ) , and Q n ( 2 ) [ u ] = h ∑ j = 1 n f ( t + j h − h / 2 ) − T 2 u ( t ) h − 1 PubDate: 2014-04-25T22:10:54Z

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): Yunxia Wei , Yanping Chen
This paper is concerned with obtaining approximate solution and approximate derivatives up to order k of the solution for neutral kth-order Volterra integro-differential equation with a regular kernel. The solution of the equation, for analytic data, is smooth on the entire interval of integration. The Legendre collocation discretization is proposed for this equation. In the present paper, we restate the initial conditions as equivalent integral equations instead of integrating two sides of the equation and provide a rigorous error analysis which justifies that not only the errors of approximate solution but also the errors of approximate derivatives up to order k of the solution decay exponentially in L 2 norm and L ∞ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
Author(s): O. Steinbach , L. Tchoualag
In this paper we describe and analyze a fast approach for the evaluation of the Newton potential for inhomogeneous partial differential equations in the particular case of two-dimensional circular domains. The method is based on suitable mesh discretization of the domain which enables to write the Newton potential in terms of matrix–vector multiplication. Moreover, this multiplication can be speed up by utilizing the fast Fourier transform (FFT) due to the circulant property of the matrices. Some numerical examples for the scalar Yukawa equation, and for the system of linear elasticity of Yukawa type which show a remarkable efficiency and the reliability of the solver are presented.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Vít Dolejší
We present a new anisotropic hp-adaptive technique, which can be employed for the numerical solution of various scientific and engineering problems governed by partial differential equations in 2D with the aid of a discontinuous piecewise polynomial approximation. This method generates anisotropic triangular grids and the corresponding polynomial approximation degrees based on the minimization of the interpolation error in the L q -norm ( q ∈ [ 1 , ∞ ] ). We develop the theoretical background of this approach and present several numerical examples demonstrating the efficiency of the anisotropic adaptive strategy.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Yong Wang , Jian-Xun Zhao
In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Yidu Yang , Lingling Sun , Hai Bi , Hao Li
In this paper we study the residual type a posteriori error estimates for general elliptic (not necessarily symmetric) eigenvalue problems. We present estimates for approximations of semisimple eigenvalues and associated eigenvectors. In particular, we obtain the following new results: 1) An error representation formula which we use to reduce the analysis of the eigenvalue problem to the analysis of the associated source problem; 2) A local lower bound for the error of an approximate finite element eigenfunction in a neighborhood of a given mesh element T.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Qiang Ma , Deqiong Ding , Xiaohua Ding
This paper focuses on mean-square dissipativity of several numerical methods applied to a class of stochastic differential equations with jumps. The conditions under which the underlying systems are mean-square dissipative are given. It is shown that the mean-square dissipativity is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce mean-square dissipativity under a stepsize constraint. Those results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of mean-square dissipativity.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Yibao Li , Junseok Kim
In this paper, we propose a new unconditionally stable hybrid numerical method for minimizing the piecewise constant Mumford–Shah functional of image segmentation. The model is based on the Allen–Cahn equation and an operator splitting technique is used to solve the model numerically. We split the governing equation into two linear equations and one nonlinear equation. One of the linear equations and the nonlinear equation are solved analytically due to the availability of closed-form solutions. The other linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical algorithm such as a multigrid method. We prove the unconditional stability of the proposed scheme. Since we incorporate closed-form solutions and an unconditionally stable scheme in the solution algorithm, our proposed scheme is accurate and robust. Various numerical results on real and synthetic images with noises are presented to demonstrate the efficiency, robustness, and accuracy of the proposed method.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Mohammed Shuker Mahmood , Karel Kovářik
The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.

Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
Author(s): Dongfang Li , Chengjian Zhang
In this paper, we investigate the convergence behavior of discontinuous Galerkin methods for solving a class of delay differential equations. Although discontinuities may occur in various orders of the derivative of the solutions, we show that the m-degree DG solutions have ( m + 1 ) th order accuracy in L ∞ norm. Numerical experiments confirm the theoretical results of the methods.

Abstract: Publication date: September 2014
Source:Applied Numerical Mathematics, Volume 83
Author(s): Hassan Majidian
A special class of weakly singular Volterra integral equations with noncompact kernels is considered. We consider a representation of the unique smooth solution of the equation and present a novel class of numerical approximations based on Gaussian quadrature rules. It is shown that the method of this type has a(n) (nearly) optimal rate of convergence under a specific condition which is very practical and easy to check. In some cases, the superconvergence property is also achieved. A stability analysis of the method is also provided. The method may be preferred to the iterated collocation method which is superconvergent under many conditions on the unknown solution. Some numerical examples are presented which are in accordance with the theoretical results.