Abstract: Publication date: Available online 3 March 2015
Source:Applied Numerical Mathematics
Author(s): Bei Zhang , Shaochun Chen , Jikun Zhao
We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction-diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.

Abstract: Publication date: January 2015
Source:Applied Numerical Mathematics, Volume 87
Author(s): J.J. Liu , M. Yamamoto , L. Yan
Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.

Abstract: Publication date: Available online 7 January 2015
Source:Applied Numerical Mathematics
Author(s): Panagiotis D. Michailidis , Konstantinos G. Margaritis
Numerical linear algebra is one of the most important forms of scientific computation. The basic computations in numerical linear algebra are matrix computations and linear systems solution. These computations are used as kernels in many computational problems. This study demonstrates the parallelisation of these scientific computations using multi-core programming frameworks. Specifically, the frameworks examined here are Pthreads, OpenMP, Intel Cilk Plus, Intel TBB, SWARM, and FastFlow. A unified and exploratory performance evaluation and a qualitative study of these frameworks are also presented for parallel scientific computations with several parameters. The OpenMP and SWARM models produce good results running in parallel with compiler optimisation when implementing matrix operations at large and medium scales, whereas the remaining models do not perform as well for some matrix operations. The qualitative results show that the OpenMP, Cilk Plus, TBB, and SWARM frameworks require minimal programming effort, whereas the other models require advanced programming skills and experience. Finally, based on an extended study, general conclusions regarding the programming models and matrix operations for some parameters were obtained.

Abstract: Publication date: Available online 7 January 2015
Source:Applied Numerical Mathematics
Author(s): Dong-wook Shin , Youngmok Jeon , Eun-Jae Park
A hybrid discontinuous Galerkin (HDG) method for the Poisson problem introduced by Jeon and Park can be viewed as a hybridizable discontinuous Galerkin method using a Baumann–Oden type local solver. In this work, an upwind HDG method with super-penalty is proposed to solve advection–diffusion–reaction problems. A super-penalty formulation facilitates an optimal order convergence in the L 2 norm as well as the energy norm. Several numerical examples are presented to show the performance of the method.

Abstract: Publication date: Available online 26 January 2015
Source:Applied Numerical Mathematics
Author(s): Yassine Boubendir , Oscar Bruno , David Levadoux , Catalin Turc
This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.

Abstract: Publication date: Available online 27 January 2015
Source:Applied Numerical Mathematics
Author(s): Nagaiah Chamakuri , Karl Kunisch , Gernot Plank
Optimal control techniques are investigated with the goal of terminating reentry waves in cardiac tissue models. In this computational study the Luo–Rudy phase-I ventricular action potential model is adopted which accounts for more biophysical details of cellular dynamics as compared to previously used phenomenological models. The parabolic and ordinary differential equations are solved as a coupled system and an AMG preconditioner is used to solve the discretized elliptic equation. The numerical results demonstrate that defibrillation is possible by delivering a single strong shock. The optimal control approach also leads to successful defibrillation and demands less total current. The present study motivates us to further investigate optimal control techniques on realistic geometries by incorporating the structural heterogeneity in the cardiac tissue.

Abstract: Publication date: February 2015
Source:Applied Numerical Mathematics, Volume 88
Author(s): Juan C. Aguilar
We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.

Abstract: Publication date: February 2015
Source:Applied Numerical Mathematics, Volume 88
Author(s): H. Temimi
In this paper, we study the superconvergence properties of the discontinuous Galerkin (DG) method applied to one-dimensional mth-order ordinary differential equations without introducing auxiliary variables. We show that the leading term of the discretization error on each element is proportional to a combination of Jacobi polynomials. Thus, the p-degree DG solution is O ( h p + 2 ) superconvergent at the roots of specific combined Jacobi polynomials. Moreover, we use these results to compute simple, efficient and asymptotically exact a posteriori error estimates and to construct higher-order DG approximations.

Abstract: Publication date: February 2015
Source:Applied Numerical Mathematics, Volume 88
Author(s): Daniel Potts , Manfred Tasche
Let h ( x ) be a nonincreasing exponential sum of order M. For N given noisy sampled values h n = h ( n ) + e n ( n = 0 , … , N − 1 ) with error terms e n , all parameters of h ( x ) can be estimated by the known ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) method. The ESPRIT method is based on singular value decomposition (SVD) of the L-trajectory matrix ( h ℓ + m ) ℓ , m = 0 L − 1 , N − L , where the window length L fulfills M ≤ L ≤ N − M + 1 . The computational cost of the ESPRIT algorithm is dominated by the cost of SVD. In the case L ≈ N 2 , the ESPRIT algorithm based on complete SVD costs about 21 8 N 3 + M 2 ( 21 N + 91 3 M ) operations. Here we show that the ESPRIT algorithm based on partial SVD and fast Hankel matrix–vector multiplications has much lower cost. Especially for L ≈ N 2 , the ESPRIT algorithm based on partial Lanczos bidiagonalization with S steps requires only about 18 S N log 2 N + S 2 ( 20 N + 30 S ) + M 2 ( N + 1 3 M ) operations, where M ≤ S ≤ N − L + 1 . Numerical experiments demonstrate the high performance of these fast ESPRIT algorithms for noisy sampled data with relatively large error terms.

Abstract: Publication date: January 2015
Source:Applied Numerical Mathematics, Volume 87
Author(s): Jaewook Lee , Younhee Lee
In this paper, we consider a local volatility model with jumps under which the price of a European option can be derived by a partial integro-differential equation (PIDE) with nonconstant coefficients. In order to solve numerically the PIDE, we generalize the implicit method with three time levels which is constructed to avoid iteration at each time step. We show that the implicit method has the stability with respect to the discrete ℓ 2 -norm by using an energy method. We combine the implicit method with an operator splitting method to solve a linear complementarity problem (LCP) with nonconstant coefficients that describes the price of an American option. Finally we conduct some numerical simulations to verify the analysis of the method. The proposed method leads to a tridiagonal linear system at each time step and thus the option prices can be computed in a few seconds on a computer.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): Fuxing Hu , Rong Wang , Xueyong Chen , Hui Feng
An adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws, which combines the rezoning approach with the finite volume weighted essentially non-oscillatory (WENO) scheme. An a posteriori error estimate, used to equidistribute the mesh, is obtained from the differences between respective numerical solutions of 5th-order WENO (WENO5) and 3rd-order ENO (ENO3) schemes. The number of grids can be adaptively readjusted based on the solution structure. For higher efficiency, mesh readjustment is performed every few time steps rather than every time step. In addition, a high order conservative interpolation is used to compute the physical solutions on the new mesh from old mesh based on the finite volume ENO reconstruction. Extensive examples suggest that this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): B. Cano , A. González-Pachón
The aim of the present paper is to study the suitability of using exponential methods for the time integration of cubic Schrödinger equation till long times. We center on second-order methods, for which we prove a higher order of accuracy on the main invariants when integrating solitary waves. Some geometric implicit exponential methods are considered as well as some explicit suitably projected ones. The comparison in terms of efficiency is performed.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): Tian-jun Wang
In this paper, we develop a direct spectral method based on the mixed Laguerre–Legendre quasi-orthogonal approximation for non-isotropic heat transfer with inhomogeneous Neumann boundary condition in an infinite strip. This method guarantees that the homogeneous boundary condition is exactly satisfied, which differs from other spectral methods for Neumann problems. For analyzing the numerical errors, some basic results on the mixed Laguerre–Legendre quasi-orthogonal approximation are established. The convergence of the proposed scheme is proved. Numerical results demonstrate the efficiency of this new approach and coincide well with the theoretical analysis.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): Mingxia Huang , Dingguo Pu
This paper describes a new algorithm for nonlinear programming with inequality constraints. The proposed approach solves a sequence of quadratic programming subproblems via the line search technique and uses a new globalization strategy. An increased flexibility in the step acceptance procedure is designed to promote long productive steps for fast convergence. Global convergence is proved under some reasonable assumptions and preliminary numerical results are presented.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): E. Berriochoa Esnaola , A. Cachafeiro López , F. Cala Rodríguez , J. Illán González , J.M. Rebollido Lorenzo
We consider the problem of evaluating ∫ − 1 1 f ( x ) G ( x ) ( 1 − x 2 ) − 1 / 2 d x , when f is smooth and G is nearly singular and non-negative. For this we construct a Gauss quadrature formula w.r.t. the weight G ( x ) ( 1 − x 2 ) − 1 / 2 . Once the factor G has been chosen, the procedure is relatively simple and mainly involves the application of FFT to compute a finite number of coefficients of the Chebyshev series expansion of G which in turn are used to calculate modified moments. It is shown that this approach is very effective when the complexity of f is high, or when f is parametric and the integral must be calculated for many values of the parameters. For this, there is presented a selection of numerical examples which allows comparison with other methods. In particular, there is considered the evaluation of Hadamard finite part integrals when the regular part of the integrand is nearly singular.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): Songhai Deng , Zhong Wan
In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms.

Abstract: Publication date: February 2015
Source:Applied Numerical Mathematics, Volume 88
Author(s): Hai-Hua Qin , Xiaodong Liu
The interior inverse scattering by an impenetrable cavity is considered. Both the sources and the measurements are placed on a curve or surface inside the cavity. As a rule of thumb, both the direct and the inverse problems suffer from interior eigenvalues. The interior eigenvalues are removed by adding an artificial obstacle with impedance boundary condition to the underlying scattering system. For this new system, we prove a reciprocity relation for the scattered field and a uniqueness theorem for the inverse problem. Some new techniques are used in the arguments of the uniqueness proof because of the Lipschitz regularity of the boundary of the cavity. The linear sampling method is used for this new scattering system for reconstructing the shape of the cavity. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the linear sampling method. In particular, the introduction of the artificial obstacle makes the linear sampling method robust to frequency.

Abstract: Publication date: February 2015
Source:Applied Numerical Mathematics, Volume 88
Author(s): Tomáš Oberhuber
The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.

Abstract: Publication date: Available online 13 February 2015
Source:Applied Numerical Mathematics
Author(s): N.S. Hoang
A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nyström (EPTRKN) methods. We proved that an s-stage FEPTRKN method has a step order p = s and a stage order r = s for any set of distinct collocation parameters ( c i ) i = 1 s . Super-convergence for the accuracy orders of these methods can be obtained if the collocation parameters ( c i ) i = 1 s satisfy some orthogonality conditions. We proved that an s-stage FEPTRKN method can attain an accuracy order p = s + 3 . Numerical experiments have shown that the new FEPTRKN methods work better than do the corresponding EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.

Abstract: Publication date: Available online 10 February 2015
Source:Applied Numerical Mathematics
Author(s): Anna Karina Fontes Gomes , Margarete Oliveira Domingues , Kai Schneider , Odim Mendes , Ralf Deiterding
We present an adaptive multiresolution method for the numerical simulation of ideal magnetohydrodynamics in two space dimensions. The discretization uses a finite volume scheme based on a Cartesian mesh and an explicit compact Runge–Kutta scheme for time integration. Harten's cell average multiresolution allows to introduce a locally refined spatial mesh while controlling the error. The incompressibility of the magnetic field is controlled by using a Generalized Lagrangian Multiplier (GLM) approach with a mixed hyperbolic-parabolic correction. Different applications to two-dimensional problems illustrate the properties of the method. For each application CPU time and memory savings are reported and numerical aspects of the method are discussed. The accuracy of the adaptive computations is assessed by comparison with reference solutions computed on a fine regular mesh.

Abstract: Publication date: March 2015
Source:Applied Numerical Mathematics, Volume 89
Author(s): M. Ganesh , T. Thompson
In this work we focus on the efficient representation and computation of the eigenvalues and eigenfunctions of the surface Schrödinger operator ( i ∇ + A 0 ) 2 that governs a class of nonlinear Ginzburg–Landau (GL) superconductivity models on rotationally symmetric Riemannian 2-manifolds S. We identify and analyze a complete orthonormal system in L 2 ( S ; C ) of eigenmodes having a variable-separated form. For the unknown functions in this ansatz, our analysis facilitates the identification of approximate spectral problems whose eigenvalues lie arbitrarily near corresponding eigenvalues of the Schrödinger operator. We then develop and implement an arbitrary order finite element method for the efficient numerical approximation of the eigenvalue problem. We also demonstrate our analysis, algorithm and its convergence rate using parallel computations performed on a variety of choices of smooth and non-smooth surfaces S.

Abstract: Publication date: March 2015
Source:Applied Numerical Mathematics, Volume 89
Author(s): V. Reshniak , A.Q.M. Khaliq , D.A. Voss , G. Zhang
We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for different noise and drift structures. The stability matrices are established in a form convenient for analyzing their impact arising from different deterministic drift integrators. Numerical examples are provided to illustrate the effectiveness and reliability of these methods.

Abstract: Publication date: March 2015
Source:Applied Numerical Mathematics, Volume 89
Author(s): Michael Dumbser , Uwe Iben , Matteo Ioriatti
In the present paper a new efficient semi-implicit finite volume method for the simulation of weakly compressible, axially symmetric flows in compliant tubes is presented. The fluid is assumed to be barotropic and a simple cavitation model is also included in the equation of state in order to model phase transition when the fluid pressure drops below the vapor pressure. The discretized flow equations lead to a mildly nonlinear system of equations that is efficiently solved with a nested Newton technique. The new numerical method has to obey only a mild CFL condition based on the flow velocity and not on the sound speed, leading to large time steps that can be used. The scheme behaves well in the presence of shock waves and phase transition, as well as in the incompressible limit. In the present approach, the radial velocity profiles and therefore the wall friction coefficient are directly computed from first principles. In the compressible regime, the new method is carefully validated against quasi-exact solutions of the Riemann problem, while it is validated against the exact solution found by Womersley for an oscillatory flow in a rigid tube in the incompressible regime.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Ming Yan , Wei Gong , Ningning Yan
We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Francesco Calabrò , Carla Manni , Francesca Pitolli
Integrals involving refinable functions are of interest in several applications ranging from discretization of PDEs to wavelet analysis. We present a procedure to construct quadrature rules with assigned nodes for these integrals. The process requires in input the refinement mask coefficients and the sequence of nodes only. The corresponding weights are computed by an iterative procedure that does not involve the solution of linear systems. The proposed approach is deeply based on the strong connection between balanced measures and integrals of refinable functions.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Jie Du , Chi-Wang Shu , Mengping Zhang
In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36,33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Franco Dassi , Bree Ettinger , Simona Perotto , Laura M. Sangalli
We present a new mesh simplification technique developed for a statistical analysis of a large data set distributed on a generic complex surface, topologically equivalent to a sphere. In particular, we focus on an application to cortical surface thickness data. The aim of this approach is to produce a simplified mesh which does not distort the original data distribution so that the statistical estimates computed over the new mesh exhibit good inferential properties. To do this, we propose an iterative technique that, for each iteration, contracts the edge of the mesh with the lowest value of a cost function. This cost function takes into account both the geometry of the surface and the distribution of the data on it. After the data are associated with the simplified mesh, they are analyzed via a spatial regression model for non-planar domains. In particular, we resort to a penalized regression method that first conformally maps the simplified cortical surface mesh into a planar region. Then, existing planar spatial smoothing techniques are extended to non-planar domains by suitably including the flattening phase. The effectiveness of the entire process is numerically demonstrated via a simulation study and an application to cortical surface thickness data.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): E. Pellegrino
This paper concerns the construction of quadrature rules based on the use of suitable refinable quasi-interpolatory operators introduced here. Convergence analysis of the obtained quadrature rules is developed and numerical examples are included.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Mengdi Zheng , Xiaoliang Wan , George Em Karniadakis
We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): Antoine Durdek , Stig Rune Jensen , Jonas Juselius , Peter Wind , Tor Flå , Luca Frediani
We have developed a new strategy to reduce the storage requirements of a multivariate function in a multiwavelet framework. We propose that alongside the commonly used adaptivity in the grid refinement one can also vary the order of the representation k as a function of the scale n. In particular the order is decreased with increasing refinement scale. The consequences of this choice, in particular with respect to the nesting of scaling spaces, are discussed and the error of the approximation introduced is analyzed. The application of this method to some examples of mono- and multivariate functions shows that our algorithm is able to yield a storage reduction up to almost 60%. In general, values between 30 and 40% can be expected for multivariate functions. Monovariate functions are less affected but are also much less critical in view of the so called “curse of dimensionality”.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): Ping Liu , Giovanni Samaey , C. William Gear , Ioannis G. Kevrekidis
In recent years, individual-based/agent-based modeling has been applied to study a wide range of applications, ranging from engineering problems to phenomena in sociology, economics and biology. Simulating such agent-based models over extended spatiotemporal domains can be prohibitively expensive due to stochasticity and the presence of multiple scales. Nevertheless, many agent-based problems exhibit smooth behavior in space and time on a macroscopic scale, suggesting that a useful coarse-grained continuum model could be obtained. For such problems, the equation-free framework [16–18] can significantly reduce the computational cost. Patch dynamics is an essential component of this framework. This scheme is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the fine-scale agent-based model in a number of small “patches”, which cover only a fraction of the spatiotemporal domain. In this work, we construct a finite-volume-inspired conservative patch dynamics scheme and apply it to a financial market agent-based model based on the work of Omurtag and Sirovich [22]. We first apply our patch dynamics scheme to a continuum approximation of the agent-based model, to study its performance and analyze its accuracy. We then apply the scheme to the agent-based model itself. Our computational experiments indicate that here, typically, the patch dynamics-based simulation needs to be performed in only 20% of the full agent simulation space, and in only 10% of the temporal domain.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Karl Kunisch , Stefan H. Reiterer
A Gautschi time-stepping scheme for optimal control of linear second order systems is proposed and analyzed. Convergence rates are proved and shown to be valid in numerical experiments. The temporal discretization is combined with finite element and spectral based spatial discretizations, which are compared among themselves.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Mingchao Cai
In this paper, several projection method based preconditioners for various incompressible flow models are studied. In the derivations of these projection method based preconditioners, we use three different types of the approximations of the inverse of the Schur complement, i.e., the exact inverse, the Cahouet–Chabard type approximation and the BFBt type approximation. We illuminate the connections and the distinctions between these projection method based preconditioners and those related preconditioners. For the preconditioners using the Cahouet–Chabard type approximation, we show that the eigenvalues of the preconditioned systems have uniform bounds independent of the parameters and most of them are equal to 1. The analysis is based on a detailed discussion of the commutator difference operator. Moreover, these results demonstrate the stability of the staggered grid discretization and reveal the effects of the boundary treatment. To further illustrate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances with those of the related preconditioners. Generalizations of the projection method based preconditioners to other saddle point problems are also discussed.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Ercília Sousa , Can Li
A one dimensional fractional diffusion model with the Riemann–Liouville fractional derivative is studied. First, a second order discretization for this derivative is presented and then an unconditionally stable weighted average finite difference method is derived. The stability of this scheme is established by von Neumann analysis. Some numerical results are shown, which demonstrate the efficiency and convergence of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Davide Buoso , Anna Karapiperi , Stefano Pozza
In this paper, we construct several sequence transformations whose kernels contain sequences of the form S n = S + a n λ n , n = 0 , 1 , … , where S and λ are unknown parameters, and ( a n ) is a known sequence. These transformations generalize Aitken's Δ 2 process. We provide certain sufficient conditions under which one of our transformations accelerates the convergence of certain types of sequences. Finally, we illustrate these theoretical results through several numerical experiments using diverging and converging sequences.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): Hailong Qiu , Liquan Mei , Hui Liu , Stephen Cartwright
In this paper, we consider a defect-correction stabilized finite element method for incompressible Navier–Stokes equations with friction boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. In the defect step, an artificial viscosity parameter σ is added to the Reynolds number as a stability factor, and the Oseen iterative scheme is applied in the correction step. H 1 × L 2 error estimations are derived for the one-step defect-correction stabilized finite element method. In the end, some numerical results are presented to verify the theoretical analysis.

Abstract: Publication date: April 2015
Source:Applied Numerical Mathematics, Volume 90
Author(s): S.J. Johnston , K. Jordaan
In this paper, we prove the quasi-orthogonality of a family of F 2 2 polynomials and several classes of F 2 3 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two F 2 3 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): Yinghong Xu , Lipu Zhang , Jing Zhang , Hua Luo
We adopt the self-adaptive strategy to update the barrier parameter of a feasible primal-dual interior-point algorithm. We obtain two adaptive updating methods, namely, cheap updates and sharp updates. We compare the effectiveness of the short updates with the adaptive update methods on some benchmark problems. The numerical results show that the sharp updates method is superior to short updates and cheap updates methods.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): Ghazala Akram , Christian Beck
In turbulent flows, local velocity differences often obey a cascade-like hierarchical dynamics, in the sense that local velocity differences at a given scale k are driven by deterministic and random forces from the next-higher scale k − 1 . Here we consider such a hierarchically coupled model with periodic boundary conditions, and show that it leads to an N-th order initial value problem, where N is the number of cascade steps. We deal in detail with the case N = 7 and introduce a non-polynomial spline method that solves the problem for arbitrary driving forces. Several examples of driving forces are considered, and estimates of the numerical precision of our method are given. We show how to optimize the numerical method to obtain a truncation error of order O ( h 5 ) rather than O ( h 2 ) , where h is the discretization step.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): Quan Zheng , Xuezheng Li , Yue Gao
In this paper, a class of hybrid difference schemes with variable weights on Bakhvalov–Shishkin mesh is proposed to compute both the solution and the derivative in quasilinear singularly perturbed convection–diffusion boundary value problems. The parameter-uniform second-order convergence of approximating to the solution and the derivative on Bakhvalov–Shishkin mesh and that of nearly second-order on Shishkin mesh are proved clearly by use of an ( l ∞ , l 1 ) -stability property, where the former sufficient conditions for uniform convergence are modestly relaxed on Bakhvalov–Shishkin mesh and are clarified on Shishkin mesh. The numerical examples support the proposed schemes with new sufficient conditions and their error estimates.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): M.I.M. Copetti , M. Aouadi
In this article, we consider a one-dimensional contact problem in generalized thermoviscoelasticity based on the Green–Lindsay theory. We prove that the energy associated to the system decays exponentially to zero and we analyze a finite element approximation. It is shown that if the continuous solution is sufficiently smooth then the error in the L 2 -norm is order of h + Δ t . Furthermore, we demonstrate that the discrete energy decays as the time increases.

Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
Author(s): Qunyan Zhou , Dan Hang
In this paper, a new nonmonotone adaptive trust region method with line search for solving unconstrained nonlinear optimization problems is introduced. The computation of the Hessian approximation is based on the usage of the weak secant equation by a diagonal definite matrix. Under some reasonable conditions, the global convergence of the proposed algorithm is established. The numerical results show the new method is effective and attractive for large scale optimization problems.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): G. Jannoun , R. Touma , F. Brock
In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.

Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
Author(s): K.R. Perline , B.T. Helenbrook
An approach for combining Arbitrary–Lagrangian–Eulerian (ALE) moving-mesh and level-set interface tracking methods is presented that allows the two methods to be used in different spatial regions and coupled across the region boundaries. The coupling allows interface shapes to be convected from the ALE method to the level-set method and vice-versa across the ALE/level-set boundary. The motivation for this is to allow high-order ALE methods to represent interface motion in regions where there is no topology change, and the level-set function to be used in regions where topology change occurs. The coupling method is based on the characteristic directions of information propagation and can be implemented in any geometrical configuration. In addition, an iterative method for the hybrid formulation has been developed that can be combined with pre-existing solution methods. Tests of a propagating interface in a uniform flow show that the hybrid approach provides accuracy equivalent to what one is able to obtain with either of the methods individually.