Applied Numerical Mathematics [9 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274 Published by Elsevier [2571 journals] [SJR: 1.208] [H-I: 44] |
- Collocation for high order differential equations with two-points Hermite
boundary conditions- Abstract: Publication date: Available online 5 October 2014
Source:Applied Numerical Mathematics
Author(s): F. Costabile , A. Napoli
For the numerical solution of high even order differential equations with two-points Hermite boundary conditions a general collocation method is derived and studied. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. An application to the solution of the beam problem is given. Numerical experiments provide favorable comparisons with other existing methods.
PubDate: 2014-10-07T15:28:37Z
- Abstract: Publication date: Available online 5 October 2014
- Fokas transform method for a brain tumor invasion model with heterogeneous
diffusion in 1+1 dimensions- Abstract: Publication date: Available online 2 October 2014
Source:Applied Numerical Mathematics
Author(s): D. Mantzavinos , M.G. Papadomanolaki , Y.G. Saridakis , A.G. Sifalakis
Gliomas are among the most aggressive forms of brain tumors. Over the last years mathematical models have been well developed to study gliomas growth. We consider a simple and well established mathematical model focused on proliferation and diffusion. Due to the heterogeneity of the brain tissue (white and grey matter) the diffusion coefficient is considered to be discontinuous. Fokas transform approach for the solution of linear PDE problems, apart from the fact that avoids solving intermediate ODE problems, yields novel integral representations of the solution in the complex plane that decay exponentially fast and converge uniformly at the boundaries. To take advantage of these properties for the solution of the model problem at hand, we have successfully implemented Fokas transform method in the multi-domain environment induced by the interface discontinuities of our problem's domain. The fact that the integral representation of the solution at any time-space point of our problem's domain is independent on any other points of the domain, except of course on initial data, coupled with a simple composite trapezoidal rule, implemented on appropriately chosen integration contours, yields a fast and efficient analytical-numerical technique capable of producing directly high-order approximations of the solution at any point of the domain requiring no prior knowledge of the solution at any other time instances or space information.
PubDate: 2014-10-07T15:28:37Z
- Abstract: Publication date: Available online 2 October 2014
- An iterative starting method to control parasitism for the Leapfrog method
- Abstract: Publication date: January 2015
Source:Applied Numerical Mathematics, Volume 87
Author(s): Terence J.T. Norton , Adrian T. Hill
The Leapfrog method is a time-symmetric multistep method, widely used to solve the Euler equations and other Hamiltonian systems, due to its low cost and geometric properties. A drawback with Leapfrog is that it suffers from parasitism. This paper describes an iterative starting method, which may be used to reduce to machine precision the size of the parasitic components in the numerical solution at the start of the computation. The severity of parasitic growth is also a function of the differential equation, the main method and the time-step. When the tendency to parasitic growth is relatively mild, computational results indicate that using this iterative starting method may significantly increase the time-scale over which parasitic effects remain acceptably small. Using an iterative starting method, Leapfrog is applied to the cubic Schrödinger equation. The computational results show that the Hamiltonian and soliton behaviour are well-preserved over long time-scales.
PubDate: 2014-10-02T14:53:31Z
- Abstract: Publication date: January 2015
- The Laguerre pseudospectral method for the radial Schrödinger
equation- Abstract: Publication date: Available online 16 September 2014
Source:Applied Numerical Mathematics
Author(s): H. Alıcı , H. Taşeli
By transforming dependent and independent variables, radial Schrödinger equation is converted into a form resembling the Laguerre differential equation. Therefore, energy eigenvalues and wavefunctions of M-dimensional radial Schrödinger equation with a wide range of isotropic potentials are obtained numerically by using Laguerre pseudospectral methods. Comparison with the results from literature shows that the method is highly competitive.
PubDate: 2014-09-18T12:22:15Z
- Abstract: Publication date: Available online 16 September 2014
- Convergence and Error Theorems for Hermite Function Pseudo-RBFs:
Interpolation on a Finite Interval by Gaussian-Localized Polynomials- Abstract: Publication date: Available online 16 September 2014
Source:Applied Numerical Mathematics
Author(s): John P. Boyd
Any basis set { ϕ j ( x ) } can be rearranged by linear combinations into a basis of cardinal functions C j ( x ) with the property that C j ( x k ) = δ j k where the x k are the interpolation points and δ j k is the usual Kronecker delta-function, equal to one when j = k and equal to zero otherwise. The interpolant to a function f ( x ) then takes the simple form f N ( x ) = ∑ j = 1 N f ( x j ) C j ( x ) . In a companion study, Boyd and Alfaro showed that the cardinal functions for five different spectrally accurate radial basis functions (RBFs) are well approximated by polynomial cardinal functions multiplied by a Gaussian function when the RBF kernels are wide and the number of interpolation points N is small or moderate. Here, we abandon RBFs by using interpolants that are Gaussian-localized polynomials. This basis is equivalent to Hermite functions, a widely used basis or unbounded domains. We prove a rigorous convergence theorem for uniform grid on a finite interval that asserts a geometric rate of convergence for such Gaussian localized polynomial interpolants. Experimentally, we show that Hermite functions are also successful for interpolation on finite irregular grids, even on random grids. If a simple formula for the construction of the cardinal basis is known, then this is great treasure: a costly dense matrix problem is unnecessary. Lagrange invented an explicit product form for polynomial cardinal functions; Hermite function cardinals can be constructed by merely multiplying Lagrange's product by a Gaussian, exp ( − q x 2 ) . We give guidelines for choosing the constant q; theory is simple because the Gaussian localizer is the same for all N cardinal functions. Gaussian RBFs are much more costly, much more ill-conditioned than Gaussian-localized polynomial interpolants.
PubDate: 2014-09-18T12:22:15Z
- Abstract: Publication date: Available online 16 September 2014
- Nonlinear PDE based numerical methods for cell tracking in zebrafish
embryogenesis- Abstract: Publication date: Available online 16 September 2014
Source:Applied Numerical Mathematics
Author(s): Karol Mikula , Róbert Špir , Michal Smíšek , Emmanuel Faure , Nadine Peyriéras
The paper presents numerical algorithms leading to an automated cell tracking and reconstruction of the cell lineage tree during the first hours of animal embryogenesis. We present results obtained for large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. Our approach consists of three basic steps - the image filtering, the cell centers detection and the cell trajectories extraction yielding the lineage tree reconstruction. In all three steps we use nonlinear partial differential equations. For the filtering the geodesic mean curvature flow in level set formulation is used, for the cell center detection the motion of level sets by a constant speed regularized by mean curvature flow is used and the solution of the eikonal equation is essential for the cell trajectories extraction. The core of our new tracking method is an original approach to cell trajectories extraction based on finding a continuous centered paths inside the spatio-temporal tree structures representing cell movement and divisions. Such paths are found by using a suitably designed distance function from cell centers detected in all time steps of the 3D+time image sequence and by a backtracking in steepest descent direction of a potential field based on this distance function. We also present efficient and naturally parallelizable discretizations of the aforementioned nonlinear PDEs and discuss properties and results of our new tracking method on artificial and real 4D data.
PubDate: 2014-09-18T12:22:15Z
- Abstract: Publication date: Available online 16 September 2014
- Method of infinite systems of equations for solving an elliptic problem in
a semistrip- Abstract: Publication date: Available online 16 September 2014
Source:Applied Numerical Mathematics
Author(s): Quang A. Dang , Dinh Hung Tran
Many problems of mechanics and physics are posed in unbounded (or infinite) domains. For solving these problems one typically limits them to bounded domains and finds ways to set appropriate conditions on artificial boundaries or use quasi-uniform grid that maps unbounded domains to bounded ones. Differently from the above methods we approach to problems in unbounded domains by infinite system of equations. Results for some one-dimensional problems are published recently. In this paper we develop this approach for an elliptic problem in an infinite semistrip. Using the idea of Polozhii in the method of summary representations we transform infinite system of three-point vector equations to infinite systems of three-point scalar equations and obtain approximate solution with a given accuracy. Numerical experiments for several examples show the effectiveness of the proposed method.
PubDate: 2014-09-18T12:22:15Z
- Abstract: Publication date: Available online 16 September 2014
- Efficient implementation of Radau collocation methods
- Abstract: Publication date: Available online 16 September 2014
Source:Applied Numerical Mathematics
Author(s): Luigi Brugnano , Felice Iavernaro , Cecilia Magherini
In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the methods, for which a splitting procedure is easily defined. The linear convergence analysis of this splitting procedure exhibits excellent properties, which are confirmed by its performance on a few numerical tests.
PubDate: 2014-09-18T12:22:15Z
- Abstract: Publication date: Available online 16 September 2014
- Superconvergent quadratic spline quasi-interpolants on Powell–Sabin
partitions- Abstract: Publication date: Available online 6 September 2014
Source:Applied Numerical Mathematics
Author(s): D. Sbibih , A. Serghini , A. Tijini
In this paper we use Normalized Powell–Sabin B-splines constructed by Dierckx [6] to introduce a new B-spline representation of Hermite Powell–Sabin interpolant of any polynomial or any piecewise polynomial over Powell–Sabin partitions of class at least C 1 in terms of their polar forms. We use this representation for constructing several superconvergent discrete quasi-interpolants. The result that we present in this paper is a generalization of the one presented in [20] with other properties.
PubDate: 2014-09-10T10:04:46Z
- Abstract: Publication date: Available online 6 September 2014
- New optimized fourth-order compact finite difference schemes for wave
propagation phenomena- Abstract: Publication date: Available online 4 September 2014
Source:Applied Numerical Mathematics
Author(s): Maurizio Venutelli
Two optimized fourth-order compact centered finite difference schemes are presented in this paper. By minimizing, over a range of the wave numbers domain, the variations of the phase speed with the wave number, an optimization least-squares problem is formulated. Hence, solving a linear algebraic system, obtained by incorporating the relations between the coefficients for the fourth-order three-parameter family schemes, the corresponding well-resolved wave number domains, and the related optimized coefficients, for two levels of accuracy, are analytically evaluated. Several dispersion comparisons, including the asymptotic behavior between the proposed and other existing optimized pentadiagonal fourth-order schemes, are presented and discussed. The schemes applicable directly on the interior nodes, are associated with a set of fourth-order boundary closure expressions. By adopting a fourth-order six-stage optimized Runge Kutta algorithm for time marching, the stability bounds, the global errors, and the computational efficiency, for the fully discrete schemes, are examined. The performances of the presented schemes, are tested on benchmark problems that involve both the one-dimensional linear convection, and the one-dimensional nonlinear shallow water equations. Finally, the one-dimensional schemes are extended to two dimensions and, using the two dimensional shallow water equations, classical applications are presented. The results allow us to propose, as the ideal candidate for simulating wave propagation problems, the scheme which corresponds to the strict level of accuracy with the maximum resolution over a narrow wave number space.
PubDate: 2014-09-10T10:04:46Z
- Abstract: Publication date: Available online 4 September 2014
- Choice of strategies for extrapolation with symmetrization in the constant
stepsize setting- Abstract: Publication date: January 2015
Source:Applied Numerical Mathematics, Volume 87
Author(s): A. Gorgey , R.P.K. Chan
Symmetrization has been shown to be efficient in solving stiff problems. In the constant stepsize setting, we study four ways of applying extrapolation with symmetrization. We observe that for stiff linear problems the symmetrized Gauss methods are more efficient than the symmetrized Lobatto IIIA methods of the same order. However, for two-dimensional nonlinear problems, the symmetrized 4-stage Lobatto IIIA method is more efficient. In all cases, we observe numerically that passive symmetrization with passive extrapolation is more efficient than active symmetrization with active extrapolation.
PubDate: 2014-09-04T08:04:33Z
- Abstract: Publication date: January 2015
- Editorial Board
- Abstract: Publication date: December 2014
Source:Applied Numerical Mathematics, Volume 86
PubDate: 2014-09-04T08:04:33Z
- Abstract: Publication date: December 2014
- Preservation of quadratic invariants of stochastic differential equations
via Runge-Kutta methods- Abstract: Publication date: Available online 27 August 2014
Source:Applied Numerical Mathematics
Author(s): Jialin Hong , Dongsheng Xu , Peng Wang
In this paper, we give conditions for stochastic Runge-Kutta (SRK) methods to preserve quadratic invariants. It is shown that SRK methods preserving quadratic invariants are symplectic. Based on both convergence order conditions and quadratic invariant-preserving conditions, we construct some SRK schemes preserving quadratic invariants with strong and weak convergence order with the help of computer algebra, respectively. Numerical experiments are executed to verify our theoretical analysis and show the superiority of these schemes.
PubDate: 2014-09-04T08:04:33Z
- Abstract: Publication date: Available online 27 August 2014
- High-order accurate difference potentials methods for parabolic problems
- Abstract: Publication date: Available online 22 August 2014
Source:Applied Numerical Mathematics
Author(s): Jason Albright , Yekaterina Epshteyn , Kyle R. Steffen
Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in [8], and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with non-matching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach.
PubDate: 2014-09-04T08:04:33Z
- Abstract: Publication date: Available online 22 August 2014
- Stability of an implicit method to evaluate option prices under local
volatility with jumps- Abstract: Publication date: Available online 24 July 2014
Source:Applied Numerical Mathematics
Author(s): Jaewook Lee , Younhee Lee
In this paper, we consider a local volatility model with jumps under which the price of a European option can be derived by a partial integro-differential equation (PIDE) with nonconstant coefficients. In order to solve numerically the PIDE, we generalize the implicit method with three time levels which is constructed to avoid iteration at each time step. We show that the implicit method has the stability with respect to the discrete ℓ 2 -norm by using an energy method. We combine the implicit method with an operator splitting method to solve a linear complementarity problem (LCP) with nonconstant coefficients that describes the price of an American option. Finally we conduct some numerical simulations to verify the analysis of the method. The proposed method leads to a tridiagonal linear system at each time step and thus the option prices can be computed in a few seconds on a computer.
PubDate: 2014-08-18T07:01:24Z
- Abstract: Publication date: Available online 24 July 2014
- A two-level higher order local projection stabilization on hexahedral
meshes- Abstract: Publication date: Available online 7 August 2014
Source:Applied Numerical Mathematics
Author(s): Lutz Tobiska
The two-level local projection stabilization with the pair ( Q r , h , Q r − 1 , 2 h disc ) , r ≥ 1 , of spaces of continuous, piecewise (mapped) polynomials of degree r on the mesh T h in each variable and discontinuous, piecewise (mapped) polynomials of degree r − 1 on the macro mesh M h in each variable satisfy a local inf-sup condition leading to optimal error estimates. In this note, we show that even the pair of spaces ( Q r , h , Q r , 2 h disc ) , r ≥ 2 , with the enriched projection space Q r , 2 h disc satisfies the local inf-sup condition and can be used in this framework. This gives a new, alternative proof of the inf-sup condition for the pair ( Q r , h , Q r − 1 , 2 h disc ) in higher order cases r ≥ 2 .
PubDate: 2014-08-18T07:01:24Z
- Abstract: Publication date: Available online 7 August 2014
- Editorial Board
- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
PubDate: 2014-08-18T07:01:24Z
- Abstract: Publication date: November 2014
- On the uniqueness and reconstruction for an inverse problem of the
fractional diffusion process- Abstract: Publication date: Available online 12 August 2014
Source:Applied Numerical Mathematics
Author(s): J.J. Liu , M. Yamamoto , L. Yan
Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.
PubDate: 2014-08-18T07:01:24Z
- Abstract: Publication date: Available online 12 August 2014
- Linearized alternating directions method for ℓ1-norm inequality
constrained ℓ1-norm minimization- Abstract: Publication date: Available online 8 July 2014
Source:Applied Numerical Mathematics
Author(s): Shuhan Cao , Yunhai Xiao , Hong Zhu
The ℓ 1 -regularization is popular in compressive sensing due to its ability to promote sparsity property. In the past few years, intensive research activities have been attracted on the algorithms for ℓ 1 -regularized least squares or its multifarious variations. In this study, we consider the ℓ 1 -norm minimization problems simultaneously with ℓ 1 -norm inequality constraints. The formulation of this problem is preferable when the measurement of a large and sparse signal is corrupted by an impulsive noise, in the mean time the noise level is given. This study proposes and investigates an inexact alternating direction method. At each iteration, as the closed-form solution of the resulting subproblem is not clear, we apply a linearized technique such that the closed-form solutions of the linearized subproblem can be easily derived. Global convergence of the proposed method is established under some appropriate assumptions. Numerical results, including comparisons with another algorithm are reported which demonstrate the superiority of the proposed algorithm. Finally, we extend the algorithm to solve ℓ 2 -norm constrained ℓ 1 -norm minimization problem, and show that the linearized technique can be avoided.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 8 July 2014
- An approximate solution of nonlinear hypersingular integral equations
- Abstract: Publication date: Available online 10 July 2014
Source:Applied Numerical Mathematics
Author(s): I.V. Boykov , E.S. Ventsel , V.A. Roudnev , A.I. Boykova
This paper describes numerical schemes based on spline-collocation method and their justifications for approximate solutions of linear and nonlinear hypersingular integral equations with singularities of the second kind. Collocations with continuous splines and piecewise constant functions are examined for solving linear hypersingular integral equations. Uniqueness of the solution has been proved. An error of approximation has been obtained for collocation with continuous spline in case a solution of equation has derivatives up to the second order. Collocation with piecewise constant functions are examined for nonlinear hypersingular equations. The convergence of the method has been justified. An estimate of error has been obtained. Illustrative examples demonstrate the accuracy and efficiency of the developed algorithms.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 10 July 2014
- A Padé compact high-order finite volume scheme for nonlinear
Schrödinger equations- Abstract: Publication date: Available online 11 July 2014
Source:Applied Numerical Mathematics
Author(s): Wei Gao , Hong Li , Yang Liu , XiaoXi Wei
In this work, a Padé compact high-order finite volume scheme is presented for the solution of one-dimensional nonlinear Schrödinger equations. The compact high-order finite volume schemes posses inherent conservation of the equations and high order accuracy within small stencils. Fourier error analysis demonstrates that the spectral resolution of the Padé compact finite volume scheme exceeds that of the standard finite volume schemes in terms of the same order of accuracy. Besides, the linear stability of the temporal disretization scheme is also performed by using the Fourier analysis. Numerical results are obtained for the nonlinear Schrödinger equations with various initial and boundary conditions, which manifests high accuracy and validity of the Padé compact finite volume scheme.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 11 July 2014
- The derivative patch interpolation recovery technique and superconvergence
for the discontinuous Galerkin method- Abstract: Publication date: Available online 14 July 2014
Source:Applied Numerical Mathematics
Author(s): Tie Zhang , Shun Yu
We consider the discontinuous Q k -finite element approximations to the elliptic boundary value problems in d-dimensional rectangular domain. A derivative recovery technique is proposed by interpolating the derivatives of discrete solution on the patch domain. Based on the superclose estimate derived in this paper, we show that the recovered derivatives possess the local and global superconvergence. Furthermore, the asymptotically exact a posteriori estimator is given on the error of gradient approximation. Finally, numerical experiments are presented to illustrate the theoretical analysis.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 14 July 2014
- Periodized radial basis functions, part I: Theory
- Abstract: Publication date: Available online 14 July 2014
Source:Applied Numerical Mathematics
Author(s): Jianping Xiao , John P. Boyd
We extend the theory of periodized RBFs. We show that the imbricate series that define the Periodic Gaussian (PGA) and Sech (PSech) basis functions are Jacobian theta functions and elliptic functions “dn”, respectively. The naive periodization fails for the Multiquadric and Inverse Multiquadric RBFs, but we are able to define periodic generalizations of these, too, by proving and exploiting a generalization of the Poisson Summation Theorem. Although applications of periodic RBFs are mostly left for another day, we do illustrate the flaws and potential by solving the Mathieu eigenproblem on both uniform and highly-adapted grids. The terms of a Fourier basis can be grouped into four classes, depending upon parity with respect to both the origin and x = π / 2 , and so, too, the Mathieu eigenfunctions. We show how to construct symmetrized periodic RBFs and illustrate these by solving the Mathieu problem using only the periodic RBFs of the same symmetry class as the targeted eigenfunctions. We also discuss the relationship between periodic RBFs and trigonometric polynomials with the aid of an explicit formula for the nonpolynomial part of the Periodic Inverse Quadratic (PIQ) basis functions. We prove that the rate of convergence for periodic RBFs is geometric, that is, the error can be bounded by exp ( − N μ ) for some positive constant μ. Lastly, we prove a new theorem that gives the periodic RBF interpolation error in Fourier coefficient space. This is applied to the “spectral-plus” question. We find that periodic RBFs are indeed sometimes orders of magnitude more accurate than trigonometric interpolation even though it has long been known that RBFs (periodic or not) reduce to the corresponding classical spectral method as the RBF shape parameter goes to 0. However, periodic RBFs are “spectral-plus” only when the shape parameter α is adaptively tuned to the particular f ( x ) being approximated and even then, only when f ( x ) satisfies a symmetry condition.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 14 July 2014
- On numerical methods for nonlinear singularly perturbed Schrödinger
problems- Abstract: Publication date: Available online 15 July 2014
Source:Applied Numerical Mathematics
Author(s): A.I. Ávila , A. Meister , M. Steigemann
Nonlinear Schrödinger equations (NSE) model several important problems in Quantum Physics and Morphogenesis. In case of singularly perturbed problems, the theory have made interesting progress, but numerical methods have not been able to come up with small values of the singular parameter ε. Moreover, the saddle-point characteristic of the associated functional is another challenge that it was first studied by Choi & McKenna, who developed the Mountain Pass Algorithm (MPA). We will focus on NSE where a uniqueness result for ground-state solutions is obtained. In this article, we develop a new method to compute positive mountain pass solutions, which improves the results for a large range of singular parameters. We extend ideas from MPA considering the singulary perturbed problems by developing a finite element approach mixed with steepest descend directions. We use a modified line search method based on Armijo's rule for improving the Newton search and Patankar trick for preserving the positiveness of the solution. To improve the range of the singular parameter, adaptive methods based on Dual Weighted Residual method are used. Our numerical experiments are performed with the deal.II library and we show that it is possible to get solutions for ε = 10 − 6 improving the current results in four order of magnitude. At this level, machine precision must be considered for further studies.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 15 July 2014
- Editorial Board
- Abstract: Publication date: October 2014
Source:Applied Numerical Mathematics, Volume 84
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: October 2014
- A cut finite element method for a Stokes interface problem
- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): Peter Hansbo , Mats G. Larson , Sara Zahedi
We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- A modified Nyström–Clenshaw–Curtis quadrature for
integral equations with piecewise smooth kernels- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): Qiong-Sheng Chen , Fu-Rong Lin
The Nyström–Clenshaw–Curtis (NCC) quadrature is a highly accurate quadrature which is suitable for integral equations with semi-smooth kernels. In this paper, we first introduce the NCC quadrature and point out that the NCC quadrature is not suitable for certain integral equation with well-behaved kernel functions such as e − t − s . We then modify the NCC quadrature to obtain a new quadrature which is suitable for integral equations with piecewise smooth kernel functions. Applications of the modified NCC quadrature to Wiener–Hopf equations and a nonlinear integral equation are presented.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- Computational methods for a mathematical model of propagation of nerve
impulses in myelinated axons- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): Pedro M. Lima , Neville J. Ford , Patricia M. Lumb
This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- Multiscale approach for stochastic elliptic equations in heterogeneous
media- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): Xin Wang , Liqun Cao , Yaushu Wong
This paper presents a multiscale analysis for stochastic elliptic equations in heterogeneous media. The main contributions are threefold: derive the convergence rate of the first-order asymptotic solution based on the periodic approximation method; develop a new technique for dealing with a large stochastic fluctuation; and present a novel multiscale asymptotic method. A multiscale finite element method is developed, and numerical results for solving stochastic elliptic equations with rapidly oscillating coefficients are reported.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- Inflow-implicit/outflow-explicit finite volume methods for solving
advection equations- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): Karol Mikula , Mario Ohlberger , Jozef Urbán
We introduce a new class of methods for solving non-stationary advection equations. The new methods are based on finite volume space discretizations and a semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. This is natural, since we know what is outflowing from a cell at the old time step but we leave the method to resolve a system of equations determined by the inflows to a cell to obtain the solution values at the new time step. The matrix of the system in our inflow-implicit/outflow-explicit (IIOE) method is determined by the inflow fluxes which results in an M-matrix yielding favorable stability properties for the scheme. Since the explicit (outflow) part is not always dominated by the implicit (inflow) part and thus some oscillations can occur, we build a stabilization based on the upstream weighted averages with coefficients determined by the flux-corrected transport approach [2,19] yielding high resolution versions, S1IIOE and S2IIOE, of the basic scheme. We prove that our new method is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We also show its formal second order accuracy in space and time for 1D advection problems with variable velocity. Although designed for non-divergence free velocity fields, we show that the basic IIOE scheme is locally mass conservative in case of divergence free velocity. Finally, we show L2-stability for divergence free velocity in 1D on periodic domains independent of the choice of the time step, and L∞-stability for the stabilized high resolution variant of the scheme. Numerical comparisons with the purely explicit schemes like the fully explicit up-wind and the Lax–Wendroff schemes were discussed in [13] and [14] where the basic IIOE was originally introduced. There it has been shown that the new scheme has good properties with respect to a balance of precision and CPU time related to a possible choice of larger time steps in our scheme. In this contribution we compare the new scheme and its stabilized variants with widely used fully implicit up-wind method. In this comparison our new schemes show better behavior with respect to stability and precision of computations for time steps several times exceeding the CFL stability condition. Our new stabilized methods are L∞ stable, second order accurate for any smooth solution and with accuracy of order 2/3 for solutions with moving discontinuities. This is opposite to implicit up-wind schemes which have accuracy order 1/2 only. All these properties hold for any choice of time step thus making our new method attractive for practical applications.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- Non-negative Matrix Factorization under equality constraints—a study
of industrial source identification- Abstract: Publication date: November 2014
Source:Applied Numerical Mathematics, Volume 85
Author(s): A. Limem , G. Delmaire , M. Puigt , G. Roussel , D. Courcot
This work is devoted to the factorization of an observation matrix into additive factors, respectively a contribution matrix G and a profile matrix F which enable to identify many pollution sources. The search for G and F is achieved through Non-negative Matrix Factorization techniques which alternatively look for the best updates on G and F. These methods are sensitive to noise and initialization, and—as for any blind source separation method—give results up to a scaling factor and a permutation. A Weighted Non-negative Matrix Factorization extension has also been proposed in the literature, so that different standard deviations of the data matrix components are taken into account. However, some estimated profile components may be inconsistent with practical experience. To prevent this issue, we propose an informed Non-negative Matrix Factorization, where some components of the profile matrix are set to zero or to a constant positive value. A special parametrization of the profile matrix is developed in order to freeze some profile components and to let free the other ones. The problem amounts to solve a family of quadratic sub-problems. A Maximization Minimization strategy leads to some global analytical expressions of both factors. These techniques are used to estimate source contributions of airborne particles from both industrial and natural influences. The relevance of the proposed approach is shown on a real dataset.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: November 2014
- Low cost a posteriori error estimators for an augmented mixed FEM in
linear elasticity- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: Available online 6 June 2014
- Parallel spectral-element direction splitting method for incompressible
Navier-Stokes equations- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: Available online 14 June 2014
- Editorial Board
- Abstract: Publication date: August 2014
Source:Applied Numerical Mathematics, Volume 82
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: August 2014
- On eigenvalue perturbation bounds for Hermitian block tridiagonal matrices
- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: September 2014
- Efficient Newton-multigrid solution techniques for higher order
space–time Galerkin discretizations of incompressible flow- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: September 2014
- Editorial Board
- Abstract: Publication date: September 2014
Source:Applied Numerical Mathematics, Volume 83
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: September 2014
- IMEX schemes for pricing options under jump–diffusion models
- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: October 2014
- High-order splitting methods for separable non-autonomous parabolic
equations- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: October 2014
- A posteriori error estimates for a discontinuous Galerkin method applied
to one-dimensional nonlinear scalar conservation laws- 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.
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: October 2014
- A modified alternating projection based prediction-correction method for
structured variational inequalities- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: Available online 18 April 2014
- A variational approach to reconstruction of an initial tsunami source
perturbation- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: Available online 19 April 2014
- Anisotropic hp-adaptive method based on interpolation error estimates in
the Lq-norm- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- An algorithm for a class of nonlinear complementarity problems with
non-Lipschitzian functions- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- A note on the residual type a posteriori error estimates for finite
element eigenpairs of nonsymmetric elliptic eigenvalue problems- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- Mean-square dissipativity of several numerical methods for stochastic
differential equations with jumps- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- An unconditionally stable hybrid method for image segmentation
- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- Solution of double nonlinear problems in porous media by a combined finite
volume–finite element algorithm- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- L∞ error estimates of discontinuous Galerkin methods for delay
differential equations- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: August 2014
- Composite quadrature rules for a class of weakly singular Volterra
integral equations with noncompact kernels- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: September 2014