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 [2563 journals] [SJR: 1.208] [H-I: 44] |
- Discontinuous Galerkin methods for fast reactive mass transfer through
semi-permeable membranes- Abstract: Publication date: Available online 25 June 2014
Source:Applied Numerical Mathematics
Author(s): Andrea Cangiani , Emmanuil H. Georgoulis , Max Jensen
A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. The case of fast reactions is also included. More specifically, a model problem consisting of a system of semilinear parabolic advection–diffusion–reaction partial differential equations in each compartment with only local Lipschitz conditions on the nonlinear reaction terms, equipped with respective initial and boundary conditions, is considered. General nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. The interior penalty dG method for this problem, presented recently, is analysed both in the space-discrete and in fully discrete settings for the case of, possibly, fast reactions. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 25 June 2014
- A two-grid method for elliptic problem with boundary layers
- Abstract: Publication date: Available online 25 June 2014
Source:Applied Numerical Mathematics
Author(s): A.I. Zadorin , S.V. Tikhovskaya , N.A. Zadorin
A two-grid method for the elliptic equation with a small parameter ε multiplying the highest derivative is investigated. The difference schemes with the property of ε-uniform convergence on a uniform mesh and on Shishkin mesh are considered. In both cases, a two-grid method for resolving the difference scheme is investigated. A two-grid method has features that are concerned with a uniform convergence of a difference scheme. To increase the accuracy, the Richardson extrapolation in two-grid method is applied. Numerical results are discussed.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 25 June 2014
- Application of transparent boundary conditions to high-order
finite-difference schemes for the wave equation in waveguides- Abstract: Publication date: Available online 5 July 2014
Source:Applied Numerical Mathematics
Author(s): I.L. Sofronov , L. Dovgilovich , N. Krasnov
We propose a method for generating finite-difference approximations of transparent boundary conditions (TBCs) with the fourth and sixth order in space. It is based on the wave equation solution continuation extra two or three layers of grid points outside the computational domain to use them in central-difference operators on approaching the boundary. We present the theoretical background of the method, give estimates of computational resources, and discuss accuracy and stability results of numerical tests in 1D and 2D cases.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 5 July 2014
- Finite element potentials
- Abstract: Publication date: Available online 8 July 2014
Source:Applied Numerical Mathematics
Author(s): Ana Alonso Rodríguez , Alberto Valli
We present an explicit and efficient way for constructing finite elements with assigned gradient, or curl, or divergence. Some simple notions of homology theory and graph theory applied to the finite element mesh are basic tools for devising the solution algorithms.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 8 July 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
- High-accuracy finite-difference schemes for solving elastodynamic problems
in curvilinear coordinates within multiblock approach- Abstract: Publication date: Available online 5 July 2014
Source:Applied Numerical Mathematics
Author(s): Leonid Dovgilovich , Ivan Sofronov
We propose highly accurate finite-difference schemes for simulating wave propagation problems described by linear second-order hyperbolic equations. The schemes are based on the summation by parts (SBP) approach modified for applications with violation of input data smoothness. In particular, we derive and implement stable schemes for solving elastodynamic anisotropic problems described by the Navier wave equation in complex geometry. To enhance potential of the method, we use a general type of coordinate transformation and multiblock grids. We also show that the conventional spectral element method (SEM) can be treated as the multiblock finite-difference method whose blocks are the SEM cells with SBP operators on GLL grid.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 5 July 2014
- Semi-implicit finite volume level set method for advective motion of
interfaces in normal direction- Abstract: Publication date: Available online 8 July 2014
Source:Applied Numerical Mathematics
Author(s): Peter Frolkovič , Karol Mikula , Jozef Urbán
In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 8 July 2014
- The mathematical modeling of the electric field in the media with
anisotropic objects- Abstract: Publication date: Available online 10 July 2014
Source:Applied Numerical Mathematics
Author(s): M.I. Epov , E.P. Shurina , N.V. Shtabel
We present a numerical scheme for modeling the electric field in the media with tensor conductivity. This scheme is based on vector finite element method in frequency domain. The numerical computations of the electric field in the anisotropic medium are done. The conductivity of the anisotropic medium is positive defined dense tensor in general case. We consider the electric field from anisotropic layer, inclined anisotropic layer and some anisotropic objects in isotropic half-space.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 10 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
- High-resolution difference methods with exact evolution for
multidimensional waves- Abstract: Publication date: Available online 10 July 2014
Source:Applied Numerical Mathematics
Author(s): Thomas Hagstrom
We consider the generalization of high-order upwind Strang methods for simulating waves. In 1 + 1 dimensions the methods can be defined via the exact evolution over a single time step of an odd-order piecewise polynomial interpolant of the grid data. We construct a true multidimensional version for acoustic waves by applying the solution operator in integral form to the interpolant. We also examine the replacement of polynomials by bandlimited interpolation functions (BLIFs). Numerical experiments with turbulent wave fields are presented to verify the accuracy and stability of the multidimensional methods and to assess the relative effectiveness of the two interpolation techniques.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 10 July 2014
- On a fictitious domain method with distributed Lagrange multiplier for
interface problems- Abstract: Publication date: Available online 11 July 2014
Source:Applied Numerical Mathematics
Author(s): Ferdinando Auricchio , Daniele Boffi , Lucia Gastaldi , Adrien Lefieux , Alessandro Reali
In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrangian multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations.
PubDate: 2014-07-18T21:55:06Z
- Abstract: Publication date: Available online 11 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
- Editorial Board
- Abstract: Publication date: July 2014
Source:Applied Numerical Mathematics, Volume 81
PubDate: 2014-06-14T16:16:58Z
- Abstract: Publication date: July 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
- Krylov subspace recycling for sequences of shifted linear systems
- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- A flux preserving immersed nonconforming finite element method for
elliptic problems- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- The numerical solution of weakly singular integral equations based on the
meshless product integration (MPI) method with error 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- Cartesian PML approximation to resonances in open systems in R2
- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- Analytic and numerical exponential asymptotic stability of nonlinear
impulsive differential equations- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- Analysis of errors in some recent numerical quadrature formulas for
periodic singular and hypersingular integrals via regularization- 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
- Legendre spectral collocation method for neutral and high-order Volterra
integro-differential equation- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 2014
- Fast Fourier transform for efficient evaluation of Newton potential in BEM
- 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.
PubDate: 2014-04-25T22:10:54Z
- Abstract: Publication date: July 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