Applied Numerical Mathematics [SJR: 1.163] [H-I: 49] [9 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274 Published by Elsevier [2766 journals] |
- Special Issue: Fourth Chilean Workshop on Numerical Analysis of Partial
Differential Equations (WONAPDE), Universidad de Concepción, Chile- Abstract: Publication date: Available online 4 May 2015
Source:Applied Numerical Mathematics
Author(s): Raimund Bürger , Gabriel N. Gatica , Norbert Heuer , Rodolfo Rodríguez , Mauricio Sepúlveda
PubDate: 2015-05-04T23:37:53Z
- Abstract: Publication date: Available online 4 May 2015
- Editorial Board
- Abstract: Publication date: May 2015
Source:Applied Numerical Mathematics, Volume 91
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: May 2015
- Functionally-fitted explicit pseudo two-step
Runge–Kutta–Nyström methods- Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
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 step order p = s and 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 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- A three-term conjugate gradient algorithm for large-scale unconstrained
optimization problems- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- Mixed spectral method for heat transfer with inhomogeneous Neumann
boundary condition in an infinite strip- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- Line search SQP method with a flexible step acceptance procedure
- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- On the acceleration of spatially distributed agent-based computations: A
patch dynamics scheme- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- Bicompact scheme for the multidimensional stationary linear transport
equation- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): E.N. Aristova , B.V. Rogov
A fourth-order accurate (in space) bicompact scheme is proposed for solving the inhomogeneous stationary transport equation in two dimensions. The scheme is based on a minimal stencil consisting of two nodes in each dimension and is obtained as a stationary limit of bicompact schemes produced by the method of lines for the nonstationary transport equation. The set of unknowns for each two-dimensional cell consists of the node values of the solution function and its integrals over cell edges and the entire cell. A closed system of linear equations is obtained for all desired variables in each cell. This system is solved using the running calculation method, which reveals the characteristic properties of the transport equation without explicitly using characteristics. The numerical results are compared with the solution produced by a conservative-characteristic method applied to a similar set of variables. The advantages of the bicompact schemes are demonstrated.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Numerical solution of stochastic differential equations in the sense of
Stratonovich in an amorphization crystal lattice model- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): G.I. Zmievskaya , T.A. Averina , A.L. Bondareva
Model phase transition /PT/ involves the formation of defects (voids or blisters), their migration into thin layers sample from SiC and Mo, accumulation of defects, and consequently, a change in the crystal lattice strain which leads to amorphization of materials. Mathematical model is related with solution of stochastic differential equations /SDEs/. The scheme used is a two-level modification of the asymptotically unbiased numerical method for solving SDEs in the sense of Stratonovich, which has second order mean-square convergence for SDEs with a single noise or for SDEs with additive noise. Based on example of computer simulation of porosity and amorphization lattice are to be discussed characteristics of phase transition at initial stage as well their influence on protective qualities of SiC thin layer cover subjected by radiation of Xe + + ions.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Editorial Board
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Professor V.S. Ryaben'kii. On the occasion of the 90-th birthday
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Yekaterina Epshteyn , Ivan Sofronov , Semyon Tsynkov
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Implementation of Nordsieck second derivative methods for stiff ODEs
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): A. Abdi , G. Hojjati
It is the purpose of this paper to study the construction and implementation of Nordsieck second derivative methods for the numerical integration of stiff systems of first order ordinary differential equations. We construct L-stable methods of order p = s + 1 , where s is the number of internal stages, and stage order q = p . The implementation issues including the starting procedures, stage predictors, local error estimation and the changing stepsize are examined. Numerical experiments with methods of orders three and four indicate reliability of the error estimates and efficiency of the methods in a variable stepsize environment.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Exact Riemann solvers for conservation laws with phase change
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Chunguang Chen , Harumi Hattori
In this paper we consider the solid–solid phase transformation in martensitic materials and present two numerical procedures for solving exactly the Riemann problems of a 3 × 3 system of conservation laws [21]. A particular attention is given to the configurations of the phase boundaries. For a Riemann problem whose initial states are specified in different phases, we first assume that the phase boundary is stationary and then find the solution through an iteration method [24]. Configuration of the transition front is then determined based on this stationary-phase-boundary solution [12]. The solution with dynamic phase change is calculated by listing all the relations in the Riemann problem and solving the resulting nonlinear system. Another approach, which avoids solving this system, is also proposed where the solution is obtained by computing the intersection of two projection curves. A front capturing/tracking method [25] implementing these Riemann solvers is presented to approximate initial value problems with propagating transition fronts. This approach captures the phase boundary sharply without artificial smearing in the physically unstable region.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Nonlinear Schwarz iterations with reduced rank extrapolation
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Sébastien Duminil , Hassane Sadok , Daniel B. Szyld
Extrapolation methods can be a very effective technique used for accelerating the convergence of vector sequences. In this paper, these methods are used to accelerate the convergence of Schwarz iterative methods for nonlinear problems. A new implementation of the reduced-rank-extrapolation (RRE) method is introduced. Some convergence analysis is presented, and it is shown numerically that certain extrapolation methods can indeed be very effective in accelerating the convergence of Schwarz methods.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- An eigenspace method for computing derivatives of semi-simple eigenvalues
and corresponding eigenvectors of quadratic eigenvalue problems- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Xin Lu , Shu-fang Xu
This paper concerns computing derivatives of semi-simple eigenvalues and corresponding eigenvectors of the quadratic matrix polynomial Q ( p , λ ) = λ 2 M ( p ) + λ C ( p ) + K ( p ) at p = p ⁎ . Computing derivatives of eigenvectors usually requires solving a certain singular linear system by transforming it into a nonsingular one. However, the coefficient matrix of the transformed linear system might be ill-conditioned. In this paper, we propose a new method for computing these derivatives, where the condition number of the coefficient matrix is the ratio of the maximum singular value to the minimum nonzero singular value of Q ( p ⁎ , λ ( p ⁎ ) ) , which is generally smaller than those in current literature and hence leads to higher accuracy. Numerical examples show the feasibility and efficiency of our method.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- On the numerical solution of a boundary integral equation for the exterior
Neumann problem on domains with corners- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): L. Fermo , C. Laurita
The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Method of lines for physiologically structured models with diffusion
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Agnieszka Bartłomiejczyk , Henryk Leszczyński
We deal with a size-structured model with diffusion. Partial differential equations are approximated by a large system of ordinary differential equations. Due to a maximum principle for this approximation method its solutions preserve positivity and boundedness. We formulate theorems on stability of the method of lines and provide suitable numerical experiments.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Convergence analysis of the summation of the factorially divergent Euler
series by Padé approximants and the delta transformation- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Riccardo Borghi , Ernst Joachim Weniger
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series E ( z ) ∼ ∑ n = 0 ∞ ( − 1 ) n n ! z n is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Padé approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Adaptive order polynomial algorithm in a multiwavelet representation
scheme- 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”.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- A new boundary element integration strategy for retarded potential
boundary integral equations- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): S. Falletta , L. Scuderi
We consider the retarded potential boundary integral equation, arising from the 3D Dirichlet exterior wave equation problem. For its numerical solution we use compactly supported temporal basis functions in time and a standard collocation method in space. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the numerical stability, we propose a new efficient and competitive quadrature strategy. We compare this approach with the one that uses the Lubich time convolution quadrature, and show pros and cons of both methods.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Bounding matrix functionals via partial global block Lanczos decomposition
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): M. Bellalij , L. Reichel , G. Rodriguez , H. Sadok
Approximations of expressions of the form I f : = trace ( W T f ( A ) W ) , where A ∈ R m × m is a large symmetric matrix, W ∈ R m × k with k ≪ m , and f is a function, can be computed without evaluating f ( A ) by applying a few steps of the global block Lanczos method to A with initial block-vector W. This yields a partial global Lanczos decomposition of A. We show that for suitable functions f upper and lower bounds for I f can be determined by exploiting the connection between the global block Lanczos method and Gauss-type quadrature rules. Our approach generalizes techniques advocated by Golub and Meurant for the standard Lanczos method (with block size one) to the global block Lanczos method. We describe applications to the computation of upper and lower bounds of the trace of f ( A ) and consider, in particular, the computation of upper and lower bounds for the Estrada index, which arises in network analysis. We also discuss an application to machine learning.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Editorial Board
- Abstract: Publication date: June 2015
Source:Applied Numerical Mathematics, Volume 92
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- Convergence of two-dimensional staggered central schemes on unstructured
triangular grids- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- A hybrid level-set/moving-mesh interface tracking method
- 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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: June 2015
- On a Numerov–Crank–Nicolson–Strang scheme with discrete
transparent boundary conditions for the Schrödinger equation on a
semi-infinite strip- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): A. Zlotnik , A. Romanova
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov–Crank–Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L 2 -stability (in particular, L 2 -conservativeness) together with the error estimate O ( τ 2 + h 4 ) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip and solving of tridiagonal systems in its main direction is developed to implement the splitting method for general potential. We also engage the Richardson extrapolation in time to increase the error order with respect to time step and get the method of higher order both in space and time. Numerical results on the tunnel effect for smooth and discontinuous rectangular barriers are included together with the careful practical error analysis on refining meshes.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Generalized convolution quadrature with variable time stepping. Part II:
Algorithm and numerical results- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Maria Lopez-Fernandez , Stefan Sauter
In this paper we address the implementation of the Generalized Convolution Quadrature (gCQ) presented and analyzed by the authors in a previous paper for solving linear parabolic and hyperbolic convolution equations. Our main goal is to overcome the current restriction to uniform time steps of Lubich's Convolution Quadrature (CQ). A major challenge for the efficient realization of the new method is the evaluation of high-order divided differences for the transfer function in a fast and stable way. Our algorithm is based on contour integral representation of the numerical solution and quadrature in the complex plane. As the main application we consider the wave equation in exterior domains, which is formulated as a retarded boundary integral equation. We provide numerical experiments to illustrate the theoretical results.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- An analysis of the Prothero–Robinson example for constructing new
adaptive ESDIRK methods of order 3 and 4- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Joachim Rang
Explicit singly-diagonally-implicit (ESDIRK) Runge–Kutta methods have usually order reduction if they are applied on stiff ODEs, such as the example of Prothero and Robinson. It can be observed that the numerical order of convergence decreases to the stage order, which is limited to two. In this paper we analyse the Prothero–Robinson example and derive new order conditions to avoid order reduction. New third and fourth order ESDIRK methods are created, which are applied to the Prothero–Robinson example and to an index-2 DAE. Numerical examples show that the new methods have better convergence properties than usual ESDIRK methods.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Stiff convergence of force-gradient operator splitting methods
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Emil Kieri
We consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Carleman estimates for the regularization of ill-posed Cauchy problems
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Michael V. Klibanov
This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- An algorithm of the method of difference potentials for domains with cuts
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): V.S. Ryaben'kii , S.V. Utyuzhnikov
The method of Difference Potentials (DPM) is applied to solving a Dirichlet problem for the Laplace equation in a square with a cut. The DPM approach has been modified to achieve a more efficient numerical algorithm with respect to computational time. The considered problem can be a prototype for other problems formulated in domains with cuts including elastic problems related to cracks.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- A method for calculating the Painlevé transcendents
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): A.A. Abramov , L.F. Yukhno
A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- A two-grid method for elliptic problem with boundary layers
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
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: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Active sound control in composite regions
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Emmanuel A. Ntumy , Sergey V. Utyuzhnikov
In active sound control, noise shielding of a target region is achieved via additional sources (called controls) situated at the perimeter of the region. The sources protect the target region by adjusting the acoustic field near the boundary of the region. In the present paper a numerical model of active sound control based on surface potentials in 3D bounded composite regions is numerically studied. In the composite region setup, it is required that the regions be shielded from noise while allowing admissible sound that is generated in the shielded regions to be preserved. The admissible sound is usually required to propagate freely inside the protected regions or in a (selective) predetermined pattern. The adjusting approach used here does not require any knowledge of the sound sources or the properties of the propagation medium in order to obtain the controls. Moreover, the approach differs sharply from some other approaches where the detailed knowledge of the sound sources and the propagation medium is required. For the first time, numerical test cases involving both free communication and predetermined communication pattern between the regions in three dimensions are considered. In all test cases, these regions are effectively shielded from the noise while any present admissible sound is preserved. In addition, selective propagation of the admissible sound between the regions is enforced. The effect of the number of controls on their operation is also studied. Whether admissible sound is present or not, the level of noise cancellation decreases linearly as fewer controls are used. In addition to the increase in size of the interference zone, the controls become individually distinguishable.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Computation of singular solutions to the Helmholtz equation with high
order accuracy- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): S. Britt , S. Petropavlovsky , S. Tsynkov , E. Turkel
Solutions to elliptic PDEs, in particular to the Helmholtz equation, become singular near the boundary if the boundary data do not possess sufficient regularity. In that case, the convergence of standard numerical approximations may slow down or cease altogether. We propose a method that maintains a high order of grid convergence even in the presence of singularities. This is accomplished by an asymptotic expansion that removes the singularities up to several leading orders, and the remaining regularized part of the solution can then be computed on the grid with the expected accuracy. The computation on the grid is rendered by a compact finite difference scheme combined with the method of difference potentials. The scheme enables fourth order accuracy on a narrow 3 × 3 stencil: it uses only one unknown variable per grid node and requires only as many boundary conditions as needed for the underlying differential equation itself. The method of difference potentials enables treatment of non-conforming boundaries on regular structured grids with no deterioration of accuracy, while the computational complexity remains comparable to that of a conventional finite difference scheme on the same grid. The method of difference potentials can be considered a generalization of the method of Calderon's operators in PDE theory. In the paper, we provide a theoretical analysis of our combined methodology and demonstrate its numerical performance on a series of tests that involve Dirichlet and Neumann boundary data with various degrees of “non-regularity”: an actual jump discontinuity, a discontinuity in the first derivative, a discontinuity in the second derivative, etc. All computations are performed on a Cartesian grid, whereas the boundary of the domain is a circle, chosen as a simple but non-conforming shape. In all cases, the proposed methodology restores the design rate of grid convergence, which is fourth order, in spite of the singularities and regardless of the fact that the boundary is not aligned with the discretization grid. Moreover, as long as the location of the singularities is known and remains fixed, a broad spectrum of problems involving different boundary conditions and/or data on “smooth” segments of the boundary can be solved economically since the discrete counterparts of Calderon's projections need to be calculated only once and then can be applied to each individual formulation at very little additional cost.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Application of transparent boundary conditions to high-order
finite-difference schemes for the wave equation in waveguides- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
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: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Finite element approximation with numerical integration for differential
eigenvalue problems- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Sergey I. Solov'ëv
Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- The mathematical modeling of the electric field in the media with
anisotropic objects- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
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: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- High-accuracy finite-difference schemes for solving elastodynamic problems
in curvilinear coordinates within multiblock approach- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
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: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Uzawa-like methods for numerical modeling of unsteady viscoplastic Bingham
medium flows- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Larisa Muravleva
The Uzawa-like algorithm is implemented for two-dimensional flows of viscoplastic fluids. The rheological model employed is the ideal Bingham model. As a test the lid-driven square-cavity benchmark problem is considered. The results for the steady-state problem are faithfully reproduced as compared to those in the literature for the shape and location of the yield surface. The proposed method is very successful at capturing both yielded and unyielded regions.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- High-order accurate monotone compact running scheme for multidimensional
hyperbolic equations- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): A.V. Chikitkin , B.V. Rogov , S.V. Utyuzhnikov
Monotone absolutely stable conservative difference schemes intended for solving quasilinear multidimensional hyperbolic equations are described. For sufficiently smooth solutions, the schemes are fourth-order accurate in each spatial direction and can be used in a wide range of local Courant numbers. The order of accuracy in time varies from the third for the smooth parts of the solution to the first near discontinuities. This is achieved by choosing special weighting coefficients that depend locally on the solution. The presented schemes are numerically efficient thanks to the simple two-diagonal (or block two-diagonal) structure of the matrix to be inverted. First the schemes are applied to system of nonlinear multidimensional conservation laws. The choice of optimal weighting coefficients for the schemes of variable order of accuracy in time and flux splitting is discussed in detail. The capabilities of the schemes are demonstrated by computing well-known two-dimensional Riemann problems for gasdynamic equations with a complex shock wave structure.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- High-resolution difference methods with exact evolution for
multidimensional waves- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
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: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Particle methods for PDEs arising in financial modeling
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Shumo Cui , Alexander Kurganov , Alexei Medovikov
We numerically study convection–diffusion equations arising in financial modeling. We focus on the convection-dominated cases, in which the diffusion coefficients are relatively small. Both finite-difference and Monte-Carlo methods which are widely used in the problems of this kind might be inefficient due to severe restrictions on the meshsize and the number of realizations needed to achieve high resolution. We propose an alternative approach based on particle methods which have extremely low numerical diffusion and thus do not have the aforementioned restrictions. Our approach is based on the operator splitting: The hyperbolic steps are made using the method of characteristics, while the parabolic steps are performed using either a special discretization of the integral representation of the solution (which leads to a deterministic particle method) or a stochastic random walk approach. We apply the designed particle methods to a variety of test problems and the numerical results indicate high accuracy, efficiency and robustness of both the deterministic and stochastic methods. In addition, our numerical experiments clearly demonstrate that the deterministic particle method outperforms its stochastic counterpart.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Stability criteria for non-self-adjoint finite differences schemes in the
subspace- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): A. Gulin
The finite differences schemes with weights for the heat conduction equation with nonlocal boundary conditions u ( 0 , t ) = 0 , γ ∂ u ∂ x ( 0 , t ) = ∂ u ∂ x ( 1 , t ) are discussed, where γ is a given real parameter. On some interval γ ∈ ( γ 1 , γ 2 ) the spectrum of the differential operator contains three eigenvalues in the left complex half-plane, while the remaining eigenvalues are located in the right half-plane. Earlier only the case of one eigenvalue λ 0 located in the left half-plane was considered. The stability criteria of finite differences schemes is formulated in the subspace induced by stable harmonics.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- High-order difference potentials methods for 1D elliptic type models
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Yekaterina Epshteyn , Spencer Phippen
Numerical approximations and modeling of many physical, biological, and biomedical problems often deal with equations with highly varying coefficients, heterogeneous models (described by different types of partial differential equations (PDEs) in different domains), and/or have to take into consideration the complex structure of the computational subdomains. The major challenge here is to design an efficient numerical method that can capture certain properties of analytical solutions in different domains/subdomains (such as positivity, different regularity/smoothness of the solutions, etc.), while handling the arbitrary geometries and complex structures of the domains. In this work, we employ one-dimensional elliptic type models as the starting point to develop and numerically test high-order accurate Difference Potentials Method (DPM) for variable coefficient elliptic problems in heterogeneous media. While the method and analysis are simple in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- High-order accurate difference potentials methods for parabolic problems
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Jason Albright , Yekaterina Epshteyn , Kyle R. Steffen
Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are crucial for the resolution of different temporal and spatial scales in many problems from physics and biology. In this paper we continue the work started in [8], and we use modest one-dimensional parabolic problems as the initial step towards the development of high-order accurate methods based on the Difference Potentials approach. The designed methods are well-suited for variable coefficient parabolic models in heterogeneous media and/or models with non-matching interfaces and with non-matching grids. Numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes. While the method and analysis are simpler in the one-dimensional settings, they illustrate and test several important ideas and capabilities of the developed approach.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Elastic collisions among peakon solutions for the Camassa–Holm
equation- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): Alina Chertock , Jian-Guo Liu , Terrance Pendleton
The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa–Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa–Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which support the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa–Holm equation under a wide class of initial data.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Effect of bulk viscosity in supersonic flow past spacecraft
- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): A.V. Chikitkin , B.V. Rogov , G.A. Tirsky , S.V. Utyuzhnikov
In this paper, we consider the effect of bulk viscosity in various hydrodynamic problems. We numerically study this effect on the front structure of the one-dimensional stationary shock wave and on the flow past blunt body. We estimate the effect of the bulk viscosity coefficient (BVC) on the heat transfer and drag of a sphere in a supersonic flow, apparently for the first time, by the numerical solution of parabolized Navier–Stokes equations. The solution is obtained by an original fast convergent method of global iterations of the longitudinal pressure gradient. The directions of further investigations of bulk viscosity are suggested.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Wave propagation in advected acoustics within a non-uniform medium under
the effect of gravity- Abstract: Publication date: July 2015
Source:Applied Numerical Mathematics, Volume 93
Author(s): S. Abarbanel , A. Ditkowski
We investigate linear wave propagation in non-uniform medium under the influence of gravity. Unlike the case of constant properties medium here the linearized Euler equations do not admit a plane-wave solution. Instead, we find a “pseudo-plane-wave”. Also, there is no dispersion relation in the usual sense. We derive explicit analytic solutions (both for acoustic and vorticity waves) which, in turn, provide some insights into wave propagation in the non-uniform case.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: July 2015
- Second-order differential equations in the Laguerre–Hahn class
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): A. Branquinho , A. Foulquié Moreno , A. Paiva , M.N. Rebocho
Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015
- Guaranteed a posteriori error estimates for nonconforming finite element
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
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.
PubDate: 2015-04-30T22:41:33Z
- Abstract: Publication date: August 2015