Abstract: Publication date: Available online 27 January 2016
Source:Applied Numerical Mathematics
Author(s): Kasra Mohaghegh, Roland Pulch, Jan ter Maten
Nowadays electronic circuits comprise about a hundred million components on slightly more than one square centimeter. The model order reduction (MOR) techniques are among the most powerful tools to conquer this complexity and scale, although the nonlinear MOR is still an open field of research. On the one hand, the MOR techniques are well developed for linear ordinary differential equations (ODEs). On the other hand, we deal with differential algebraic equations (DAEs), which result from models based on network approaches. There are the direct and the indirect strategy to convert a DAE into an ODE. We apply the direct approach, where an artificial parameter is introduced in the linear system of DAEs. This results in a singular perturbed problem. On compact domains, uniform convergence of the transfer function of the regularized system towards the transfer function of the system of DAEs is proved in the general linear case. This convergence is for the transfer functions of the full model. We apply and investigate two different ways of MOR techniques in this context. We have two test examples, which are both TL models.

Abstract: Publication date: Available online 21 January 2016
Source:Applied Numerical Mathematics
Author(s): Farshid Dabaghi, Adrien Petrov, Jérôme Pousin, Yves Renard
This paper deals with a one-dimensional elastodynamic contact problem and aims to highlight some new numerical results. A new proof of existence and uniqueness results is proposed. More precisely, the problem is reformulated as a differential inclusion problem, the existence result follows from some a priori estimates obtained for the regularized problem while the uniqueness result comes from a monotonicity argument. An approximation of this evolutionary problem combining the finite element method as well as the mass redistribution method which consists on a redistribution of the body mass such that there is no inertia at the contact node, is introduced. Then two benchmark problems, one being new with convenient regularity properties, together with their analytical solutions are presented and some possible discretizations using different time-integration schemes are described. Finally, numerical experiments are reported and analyzed.

Abstract: Publication date: Available online 15 January 2016
Source:Applied Numerical Mathematics
Author(s): M.J. Ruijter, C.W. Oosterlee
We develop a Fourier method to solve quite general backward stochastic differential equations (BSDEs) with second-order accuracy. The underlying forward stochastic differential equation (FSDE) is approximated by different Taylor schemes, such as the Euler, Milstein, and Order 2.0 weak Taylor schemes, or by exact simulation. A θ-time-discretization of the time-integrands leads to an induction scheme with conditional expectations. The computation of the conditional expectations appearing relies on the availability of the characteristic function for these schemes. We will use the characteristic function of the discrete forward process. The expected values are approximated by Fourier cosine series expansions. Numerical experiments show rapid convergence of our efficient probabilistic numerical method. Second-order accuracy is observed and also proved. We apply the method to, among others, option pricing problems under the Constant Elasticity of Variance and Cox-Ingersoll-Ross processes.

Abstract: Publication date: Available online 11 January 2016
Source:Applied Numerical Mathematics
Author(s): Subhashree Mohapatra, Akhlaq Husain
In this paper we propose a non-conforming least squares spectral element method for Stokes equations on three dimensional domains. Any kind of first order transformation has been avoided by using a block diagonal preconditioner for velocity and pressure variables. Preconditioned conjugate gradient method is used to obtain the numerical solution. The method is proved to be exponentially accurate. Numerical results are provided to validate the proposed estimates.

Abstract: Publication date: Available online 11 January 2016
Source:Applied Numerical Mathematics
Author(s): Kuan Xu
In the last thirty years, the Chebyshev points of the first kind have not been given as much attention for numerical applications as the second-kind ones. This survey summarizes theorems and algorithms for first-kind Chebyshev points with references to the existing literature. Benefits from using the first-kind Chebyshev points in various contexts are also discussed.

Abstract: Publication date: Available online 8 January 2016
Source:Applied Numerical Mathematics
Author(s): C. Reisinger, P.A. Forsyth
An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi–Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases.

Abstract: Publication date: Available online 31 December 2015
Source:Applied Numerical Mathematics
Author(s): Stefano Giani, Luka Grubišić, Jeffrey S. Ovall
We present an hp-adaptive continuous Galerkin (hp-CG) method for approximating eigenvalues of elliptic operators, and demonstrate its utility on a collection of benchmark problems having features seen in many important practical applications—for example, high-contrast discontinuous coefficients giving rise to eigenfunctions with reduced regularity. In this continuation of our benchmark study, we concentrate on providing reliability estimates for assessing eigenfunction/invariant subspace error. In particular, we use these estimates to justify the observed robustness of eigenvalue error estimates in the presence of repeated or clustered eigenvalues. We also indicate a means for obtaining efficiency estimates from the available efficiency estimates for the associated boundary value (source) problem. As in the first part of the paper we provide extensive numerical tests for comparison with other high-order methods and also extend the list of analyzed benchmark problems.

Abstract: Publication date: Available online 2 December 2015
Source:Applied Numerical Mathematics
Author(s): Christian Hendricks, Matthias Ehrhardt, Michael Günther
In this article we combine the ideas of high-order (HO) and alternating direction implicit (ADI) schemes on sparse grids for diffusion equations with mixed derivatives. With the help of HO and ADI schemes solutions can be computed, which are fourth-order accurate in space and second-order accurate in time. For each implicit step of the ADI scheme we use a high-order-compact (HOC) discretisation such that the computational effort consists of only solving tridiagonal systems. In order to reduce the number of grid points, we use the combination technique to construct a solution defined on the sparse grid. This approach allows to further reduce the computational effort and memory consumption.

Abstract: Publication date: Available online 30 November 2015
Source:Applied Numerical Mathematics
Author(s): Chokri Chniti, Sharefa Eisa Ali Alhazmi, Sami Eltoum, Moncef Toujani
The aim of this paper is to derive and evaluate new approximations of the Dirichlet-to-Neumann (DtN) and Neumann-to-Dirichlet (NtD) maps for two-dimensional acoustic scattering problems. Some formal approximations for the two-dimensional case are derived. These various approximations are next numerically validated and compared.

Abstract: Publication date: Available online 19 November 2015
Source:Applied Numerical Mathematics
Author(s): Alexandru Mihai Bica
In this paper we propose a new iterative numerical method for initial value problems of first and second order involving retarded argument. The method uses a quadratic spline interpolation procedure activated at each iterative step. The convergence of this method of iterated splines is theoretically proven and tested on some numerical examples.

Abstract: Publication date: Available online 19 November 2015
Source:Applied Numerical Mathematics
Author(s): David P. Nicholls, Venu Tammali
The accurate simulation of linear electromagnetic scattering by diffraction gratings is crucial in many technologies of scientific and engineering interest. In this contribution we describe a High–Order Perturbation of Surfaces (HOPS) algorithm built upon a class of Integral Equations due to the analysis of Fokas and collaborators, now widely known as the Unified Transform Method. The unknowns in this formalism are boundary quantities (the electric field and current at the layer interface) which are an order of magnitude fewer than standard volumetric approaches such as Finite Differences and Finite Elements. With detailed numerical experiments we show the efficiency, fidelity, and high–order accuracy one can achieve with an implementation of this algorithm.

Abstract: Publication date: Available online 12 November 2015
Source:Applied Numerical Mathematics
Author(s): Christophe De Luigi, Jérôme Lelong, Sylvain Maire
We improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black-Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten.

Abstract: Publication date: Available online 9 October 2015
Source:Applied Numerical Mathematics
Author(s): S.A. Hosseini, A. Abdi
This paper deals with the stability analysis of the composite barycentric rational quadrature method (CBRQM) for the second kind Volterra integral equations through application to the standard and the convolution test equations. In each case, some theoretical results are achieved by providing corresponding recurrence relation and stability matrix. Verification of these theoretical results is obtained by some numerical experiments.

Abstract: Publication date: Available online 25 September 2015
Source:Applied Numerical Mathematics
Author(s): Harald Garcke, Michael Hinze, Christian Kahle
A new time discretization scheme for the numerical simulation of two-phase flow governed by a thermodynamically consistent diffuse interface model is presented. The scheme is consistent in the sense that it allows for a discrete in time energy inequality. An adaptive spatial discretization is proposed that conserves the energy inequality in the fully discrete setting by applying a suitable post processing step to the adaptive cycle. For the fully discrete scheme a quasi-reliable error estimator is derived which estimates the error both of the flow velocity, and of the phase field. The validity of the energy inequality in the fully discrete setting is numerically investigated.

Abstract: Publication date: December 2015
Source:Applied Numerical Mathematics, Volume 98
Author(s): Slimane Adjerid, Nabil Chaabane
In this manuscript we present an error analysis for the local discontinuous Galerkin method for a model elliptic problem on Cartesian meshes when polynomials of degree at most k and an appropriate approximation of the boundary condition are used. This special approximation allows us to achieve k + 1 order of convergence for both the potential and its gradient in the L 2 norm. Here we improve on existing estimates for the solution gradient by a factor h .

Abstract: Publication date: Available online 16 September 2015
Source:Applied Numerical Mathematics
Author(s): Pietro Dell'Acqua, Claudio Estatico
An acceleration technique for multiplicative iterative methods, such as Lucy-Richardson and Image Space Reconstruction Algorithm, is presented. The technique is inspired by the Landweber method in Banach spaces and is based on the application of duality maps, which allow to compute the iterations in the dual space. We show the link between the proposed acceleration and the previously known Meinel acceleration, which consists in the introduction of an exponent in the basic iterative formulas. We prove that the new acceleration technique is more stable than the Meinel acceleration. This implies that, in the restoration process, the former is able to get better accuracy and higher speeding-up than the latter. In addition to the main focus of the paper, we propose a generalization of the Landweber method in Banach space, in order to overcome some drawbacks of this recent strategy when compared with classical (Hilbertian) Landweber method. Numerical results show the behaviour and the features of the several techniques considered, highlighting the goodness of our proposals.

Abstract: Publication date: January 2016
Source:Applied Numerical Mathematics, Volume 99
Author(s): F. Guillén-González, J.R. Rodríguez Galván
The stability of velocity and pressure mixed approximations of the Stokes problem is studied, when different finite element (FE) spaces for each component of the velocity field are considered. Using the macro-element technique of Stenberg, analytical results are obtained for some new combinations of FE with globally continuous and piecewise linear pressure. These new combinations are introduced with the idea of reducing the number of degrees of freedom in some of the velocity components. Although the resulting FE are not stable in general, we show their stability in a wide family of meshes (uniformly unstructured meshes). Moreover, this method can be extended to any mesh family whenever a post-processing be performed in order to convert it in an unstructured mesh. Some 2D and 3D numerical simulations are provided to agree with the previous analysis.

Abstract: Publication date: Available online 15 September 2015
Source:Applied Numerical Mathematics
Author(s): Yanlai Chen
We present a certified version of the Natural-Norm Successive Constraint Method (cNNSCM) for fast and accurate Inf-Sup lower bound evaluation of parametric operators. Successive Constraint Methods (SCM) are essential tools for the construction of a lower bound for the inf-sup stability constants which are required in a posteriori error analysis of reduced basis approximations. They utilize a Linear Program (LP) relaxation scheme incorporating continuity and stability constraints. The natural-norm approach linearizes a lower bound of the inf-sup constant as a function of the parameter. The Natural-Norm Successive Constraint Method (NNSCM) combines these two aspects. It uses a greedy algorithm to select SCM control points which adaptively construct an optimal decomposition of the parameter domain, and then apply the SCM on each domain. Unfortunately, the NNSCM produces no guarantee for the quality of the lower bound. Through multiple rounds of optimal decomposition, the new cNNSCM provides an upper bound in addition to the lower bound and let the user control the gap, thus the quality of the lower bound. The efficacy and accuracy of the new method is validated by numerical experiments.

Abstract: Publication date: Available online 9 September 2015
Source:Applied Numerical Mathematics
Author(s): M.S. Hussein, D. Lesnic, M.I. Ivanchov, H.A. Snitko
Multiple time-dependent coefficient identification thermal problems with an unknown free boundary are investigated. The difficulty in solving such inverse and ill-posed free boundary problems is amplified by the fact that several quantities of physical interest (conduction, convection/advection and reaction coefficients) have to be simultaneously identified. The additional measurements which render a unique solution are given by the heat moments of various orders together with a Stefan boundary condition on the unknown moving boundary. Existence and uniqueness theorems are provided. The nonlinear and ill-posed problems are numerically discretised using the finite-difference method and the resulting system of equations is solved numerically using the MATLAB toolbox routine l s q n o n l i n applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerically obtained results from some typical test examples are presented and discussed in order to illustrate the efficiency of the computational methodology adopted.

Abstract: Publication date: Available online 9 September 2015
Source:Applied Numerical Mathematics
Author(s): Lyonell Boulton
The Galerkin method can fail dramatically when applied to eigenvalues in gaps of the extended essential spectrum. This phenomenon, called spectral pollution, is notoriously difficult to predict and it can occur in models from relativistic quantum mechanics, solid state physics, magnetohydrodynamics and elasticity theory. The purpose of this survey paper is two-folded. On the one hand, it describes a rigourous mathematical framework for spectral pollution. On the other hand, it gives an account on two complementary state-of-the-art Galerkin-type methods for eigenvalue computation which prevent spectral pollution completely.

Abstract: Publication date: Available online 11 September 2015
Source:Applied Numerical Mathematics
Author(s): Xiaolin Li
The moving least square (MLS) approximation is one of the most important methods to construct approximation functions in meshless methods. For the error analysis of the MLS-based meshless methods it is fundamental to have error estimates of the MLS approximation in the generic n-dimensional Sobolev spaces. In this paper, error estimates for the MLS approximation are obtained in the W k , p norm in arbitrary n dimensions when weight functions satisfy certain conditions. The element-free Galerkin (EFG) method is a typical Galerkin method combined with the use of the MLS approximation. The error results of the MLS approximation are then used to yield error estimates of the EFG method for solving both Neumann and Dirichlet boundary value problems. Finally, some numerical examples are given to confirm the theoretical analysis.

Abstract: Publication date: Available online 11 September 2015
Source:Applied Numerical Mathematics
Author(s): A. Napoli
In this paper we use Bernoulli polynomials to derive a new spectral method to find the numerical solutions of second order linear initial value problems. Stability and error analysis of this method are studied. Numerical examples are presented which support theoretical results and provide favorable comparisons with other existing methods.

Abstract: Publication date: Available online 11 September 2015
Source:Applied Numerical Mathematics
Author(s): G. Ebadi, N. Alipour, C. Vuik
Global Krylov subspace methods are among the most efficient algorithms to solve matrix equation A X = B . Deflation and augmentation techniques are used to accelerate the convergence of Krylov subspace methods. There are two different approaches for deflated and augmented methods: an augmentation space is applied explicitly in every step, or the global method is used for solving a projected problem and then a correction step is applied at the end. In this paper, we present a framework of deflation and augmentation approaches for accelerating the convergence of the global methods for the solution of nonsingular linear matrix equations A X = B . Then, we define deflated and augmented global algorithms. Also, we analyze the deflated and augmented global minimal residual and global orthogonal residual methods. Finally, we present numerical examples to illustrate the effectiveness of different versions of the new algorithms.

Abstract: Publication date: Available online 24 August 2015
Source:Applied Numerical Mathematics
Author(s): Ninoslav Truhar, Suzana Miodragović
In this paper, new relative perturbation bounds for the eigenvalues as well as for the eigensubspaces are developed for definite Hermitian matrix pairs and the quadratic hyperbolic eigenvalue problem. First, we derive relative perturbation bounds for the eigenvalues and the sin Θ type theorems for the eigensubspaces of the definite matrix pairs ( A , B ) , where both A , B ∈ C m × m are Hermitian nonsingular matrices with particular emphasis, where B is a diagonal of ±1. Further, we consider the following quadratic hyperbolic eigenvalue problem ( μ 2 M + μ C + K ) x = 0 , where M , C , K ∈ C n × n are given Hermitian matrices. Using proper linearization and new relative perturbation bounds for definite matrix pairs ( A , B ) , we develop corresponding relative perturbation bounds for the eigenvalues and the sin Θ type theorems for the eigensubspaces for the considered quadratic hyperbolic eigenvalue problem. The new bounds are uniform and depend only on matrices M, C, K, perturbations δM, δC and δK and standard relative gaps. The quality of new bounds is illustrated through numerical examples.

Abstract: Publication date: Available online 14 August 2015
Source:Applied Numerical Mathematics
Author(s): E. O'Riordan, J. Quinn
A linear time dependent singularly perturbed convection-diffusion problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.

Abstract: Publication date: Available online 12 August 2015
Source:Applied Numerical Mathematics
Author(s): Xiaojie Wang, Siqing Gan, Desheng Wang
In this paper, mean-square convergence and mean-square stability of θ-Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 1 2 , and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h > 0 . Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ-Maruyama method with θ = 1 can preserve the exponential mean-square stability for any step size. Additionally, the θ-Maruyama method with 1 2 ≤ θ ≤ 1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results.

Abstract: Publication date: Available online 13 August 2015
Source:Applied Numerical Mathematics
Author(s): Jorge Tiago
We deal with the numerical approximation of the problem of local stabilization of Burgers equation. We consider the case when only partial boundary measurements are available. An estimator is coupled with a feedback law in order to stabilize the discretized system. Two different feedback laws are compared. Their performance is analyzed in different domains related to idealized cardiovascular geometries, with increasing complexity.

Abstract: Publication date: Available online 14 August 2015
Source:Applied Numerical Mathematics
Author(s): Dinh-Liem Nguyen
This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.

Abstract: Publication date: Available online 8 August 2015
Source:Applied Numerical Mathematics
Author(s): J.F. Wang, F.X. Sun, Y.M. Cheng, A.X. Huang
In this paper, the interpolating moving least-squares (IMLS) method is discussed in details. A simpler expression of the approximation function of the IMLS method is obtained. Compared with the moving least-squares (MLS) approximation, the shape function of the IMLS method satisfies the property of Kronecker δ function. Then the meshless method based on the IMLS method can overcome the difficulties of applying the essential boundary conditions. The error estimates of the approximation function and its first and second order derivatives of the IMLS method are presented in n-dimensional space. The theoretical results show that if the weight function is sufficiently smooth and the order of the polynomial basis functions is big enough, the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes. Then the interpolating element-free Galerkin (IEFG) method based on the IMLS method is presented for potential problems. The advantage of the IEFG method is that the essential boundary conditions can be applied directly and easily. For the purpose of demonstration, some selected numerical examples are given to prove the theories in this paper.

Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): R. Touma, C. Klingenberg
We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non-zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.

Abstract: Publication date: Available online 5 August 2015
Source:Applied Numerical Mathematics
Author(s): Bernd Kugelmann, Roland Pulch
The numerical simulation of electric circuits including multirate signals can be done by a model based on partial differential algebraic equations. In the case of frequency modulated signals, a local frequency function appears as a degree of freedom in the model. Thus the determination of a solution with a minimum amount of variation is feasible, which allows for resolving on relatively coarse grids. We prove the existence and uniqueness of the optimal solutions in the case of initial-boundary value problems as well as biperiodic boundary value problems. The minimisation problems are also investigated and interpreted in the context of optimal control. Furthermore, we construct a method of characteristics for the computation of optimal solutions in biperiodic problems. Numerical simulations of test examples are presented.

Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): Francesc Aràndiga, José Jaime Noguera
We develop a reconstruction that combines interpolation and least squares fitting for point values in the context of multiresolution a la Harten. We study the smoothness properties of the reconstruction as well as its approximation order. We analyze how different adaptive techniques (ENO, SR and WENO) can be used within this reconstruction. We present some numerical examples where we compare the results obtained with the classical interpolation and the interpolation combined with least-squares approximation.

Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): Wenting Chen, Xiang Xu, Song-Ping Zhu
This paper investigates the pricing of American options under the finite moment log-stable (FMLS) model. Under the FMLS model, the price of American-style options is governed by a highly nonlinear fractional partial differential equation (FPDE) system, which is much more complicated to solve than the corresponding Black–Scholes (B–S) system, with difficulties arising from the semi-globalness of the fractional operator, in conjunction with the nonlinearity associated with the early exercise nature of American-style options. Albeit difficult, in this paper, we propose a new predictor–corrector scheme based on the spectral-collocation method to solve for the prices of American options under the FMLS model. In the current approach, the nonlinearity of the pricing system is successfully dealt with using the predictor–corrector framework, whereas the non-localness of the fractional operator is elegantly handled. We have also provided an elegant error analysis for the current approach. Various numerical experiments suggest that the current method is fast and efficient, and can be easily extended to price American-style options under other fractional diffusion models. Based on the numerical results, we have also examined quantitatively the influence of the tail index on American put options.

Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): Alemdar Hasanov, Balgaisha Mukanova
In this study, we investigate the inverse problem of identifying an unknown spacewise-dependent source F ( x ) in the one-dimensional advection–diffusion equation u t = Du xx − vu x + F ( x ) H ( t ) , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ] , based on boundary concentration measurements g ( t ) : = u ( ℓ , t ) . Most studies have attempted to reconstruct an unknown spacewise-dependent source F ( x ) from the final observation u T ( x ) : = u ( x , T ) , but from an engineering viewpoint, the above boundary data measurements are feasible. Thus, we propose a new algorithm for reconstructing the spacewise-dependent source F ( x ) . This algorithm is based on Fourier expansion of the direct problem solution followed by minimization of the cost functional by taking a partial K-sum of the Fourier expansion. Tikhonov regularization is then applied to the ill-posed problem that is obtained. The proposed approach also allows us to estimate the degree of ill-posedness for the inverse problem considered in this study. We then establish the relationship between the noise level γ > 0 , the parameter of regularization α > 0 , and the truncation (or cut-off) parameter K. A new numerical filtering algorithm is proposed for smoothing the noisy output data. Our numerical results demonstrated that the results obtained for random noisy data up to noise levels of 7% had sufficiently high accuracy for all reconstructions.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Youngjoon Hong , Djoko Wirosoetisno
It is well known that the (exact) solutions of the 3d Navier–Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity. They also remain bounded for a finite time for arbitrary initial data in L 2 . In this article, we consider two temporal discretisations (semi-implicit and fully implicit) of the 3d Navier–Stokes equations in a periodic domain and prove that their solutions remain uniformly bounded in H 1 subject to essentially the same respective smallness conditions as the continuous system (on initial data and forcing or on the time of existence) provided the time step is small.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Dingwen Deng
In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L 2 - and H 1 -norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Bouchra Bensiali , Guillaume Chiavassa , Jacques Liandrat
This paper is devoted to the mathematical analysis of a method based on fictitious domain approach. Boundary conditions of Robin type (also known as Fourier boundary conditions) are enforced using a penalization method. A complete description of the method and a full analysis are provided for univariate elliptic and parabolic problems using finite difference approximation. Numerical evidence of the predicted estimations is provided as well as numerical results for a nonlinear problem and a first extension of the method in the bivariate situation is proposed.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Hans-Goerg Roos , Ljiljana Teofanov , Zorica Uzelac
In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Arthur A. Mitrano , Rodrigo B. Platte
Approximation properties of divergence-free vector fields by global and local solenoidal bases are studied. A comparison between interpolants generated with radial kernels and multivariate polynomials is presented. Numerical results show higher rates of convergence for derivatives of the vector field being approximated in directions enforced by the divergence operator when a rectangular grid is used. We also compute the growth of Lebesgue constants for uniform and clustered nodes and study the flat limit of divergence-free interpolants based on radial kernels. Numerical results are presented for two- and three-dimensional vector fields.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Samir Karaa , Amiya K. Pani , Sangita Yadav
An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz–Volterra projection, a priori hp-error estimates in L ∞ ( L 2 ) -norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L ∞ ( L 2 ) -norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Ron Goldman , Plamen Simeonov
The ( q , h ) -Bernstein–Bézier curves are generalizations of both the h-Bernstein–Bézier curves and the q-Bernstein–Bézier curves. We investigate two essential features of ( q , h ) -Bernstein bases and ( q , h ) -Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the ( q , h ) -Bernstein bases for a non-empty interval [ a , b ] satisfy Descartes' law of signs on [ a , b ] when q > − 1 , q ≠ 0 , and h ≤ min { ( 1 − q ) a , ( 1 − q ) b } . We conclude that the corresponding ( q , h ) -Bézier curves are variation diminishing. We also derive a degree elevation formula for ( q , h ) -Bernstein bases and ( q , h ) -Bézier curves over arbitrary intervals [ a , b ] . We show that these degree elevation formulas depend only on the parameter q and are independent of both the parameter h and the interval [ a , b ] . We investigate the convergence of the control polygons generated by repeated degree elevation. We show that unlike classical Bézier curves, the control polygons generated by repeated degree elevation for ( q , h ) -Bézier curves with 0 < q < 1 do not converge to the original ( q , h ) -Bézier curve, but rather to a piecewise linear curve with vertices that depend only on q and the monomial coefficients of the q-Bézier curve having the same control points as the original ( q , h ) -Bézier curve. A similar result holds when q > 1 . Here the control polygons generated by repeated degree elevation converge to a piecewise linear curve that depends only on q and the monomial coefficients of the 1 / q -Bézier curve with the control points of the original ( q , h ) -Bézier curve in reverse order.

Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Dominique Küpper , Anne Kværnø , Andreas Rößler
The problem of solving stochastic differential-algebraic equations (SDAEs) of index 1 with a scalar driving Wiener process is considered. Recently, the authors have proposed a class of stiffly accurate stochastic Runge–Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, classifications for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability of the considered class of SRK methods is analyzed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.

Abstract: Publication date: Available online 14 May 2015
Source:Applied Numerical Mathematics
Author(s): R. Andreev , O. Scherzer , W. Zulehner
We consider the simultaneous estimation of an optical flow field and an illumination source term in a movie sequence. The particular optical flow equation is obtained by assuming that the image intensity is a conserved quantity up to possible sources and sinks which represent varying illumination. We formulate this problem as an energy minimization problem and propose a space-time simultaneous discretization for the optimality system in saddle-point form. We investigate a preconditioning strategy that renders the discrete system well-conditioned uniformly in the discretization resolution. Numerical experiments complement the theory.

Abstract: Publication date: Available online 29 April 2015
Source:Applied Numerical Mathematics
Author(s): N. Tuncer , A. Madzvamuse , A.J. Meir
In this paper we present a robust, efficient and accurate finite element method for solving reaction-diffusion systems on stationary spheroidal surfaces (these are surfaces which are deformations of the sphere such as ellipsoids, dumbbells, and heart-shaped surfaces) with potential applications in the areas of developmental biology, cancer research, wound healing, tissue regeneration, and cell motility among many others where such models are routinely used. Our method is inspired by the radially projected finite element method introduced in [39]. (Hence the name “projected” finite element method). The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. To demonstrate the robustness, applicability and generality of this numerical method, we present solutions of reaction-diffusion systems on various spheroidal surfaces. We show that the method presented in this paper preserves positivity of the solutions of reaction-diffusion equations which is not generally true for Galerkin type methods. We conclude that surface geometry plays a pivotal role in pattern formation. For a fixed set of model parameter values, different surfaces give rise to different pattern generation sequences of either spots or stripes or a combination (observed as circular spot-stripe patterns). These results clearly demonstrate the need for detailed theoretical analytical studies to elucidate how surface geometry and curvature influence pattern formation on complex surfaces.