Authors:Miloud Sadkane; Roger B. Sidje Pages: 609 - 628 Abstract: Although the principle of alternating maximization is well known in the optimization literature, it has not been used before in the context of calculating the hump of the matrix exponential. We propose a method that applies alternating maximization in this particular context, and we show that it has a number of advantages over traditional Newton-like methods. We establish convergence results that fit this context with mild assumptions than would otherwise be the case in general optimization problems. We conduct numerical tests to complement the theory and they show convergence in just a few iterations. PubDate: 2017-09-01 DOI: 10.1007/s10543-016-0644-7 Issue No:Vol. 57, No. 3 (2017)

Authors:Helle Hallik; Peeter Oja Pages: 629 - 648 Abstract: The histopolation with quadratic/linear rational splines of class \(C^2\) is studied. Such kind of splines keep the sign of its second derivative on the whole interval and, consequently, the given histogram should be strictly convex or strictly concave. The grid points of the histogram and suitable number of the spline knots between them are supposed to place arbitrarily. The uniqueness of such an histopolant is established. It is shown that the histopolant may not exist but some sufficient conditions for the existence are given. Presented numerical results confirm their adequacy. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0645-1 Issue No:Vol. 57, No. 3 (2017)

Authors:Xiaofei Zhao Pages: 649 - 683 Abstract: We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0646-0 Issue No:Vol. 57, No. 3 (2017)

Authors:S. Yeganeh; R. Mokhtari; J. S. Hesthaven Pages: 685 - 707 Abstract: This paper is devoted to determining a space-dependent source term in an inverse problem of the time-fractional diffusion equation. We use a method based on a finite difference scheme in time and a local discontinuous Galerkin method in space and investigate the numerical stability and convergence of the proposed method. Finally, various numerical examples are used illustrate the effectiveness and accuracy of the method. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0648-y Issue No:Vol. 57, No. 3 (2017)

Authors:Xingjun Luo; Chunmei Zeng; Suhang Yang; Rong Zhang Pages: 709 - 730 Abstract: Multiscale collocation methods are developed for solving ill-posed Fredholm integral equations of the first kind in Banach spaces, if the associated resolvent integral operator fulfils a condition with respect to a interval. We apply a multiscale collocation method with a matrix compression strategy to discretize the integral equation of the second kind obtained by using the Lavrentiev regularization from the original ill-posed integral equation and then use the multilevel augmentation method to solve the resulting discrete equation. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0649-x Issue No:Vol. 57, No. 3 (2017)

Authors:Guang-Da Hu; Taketomo Mitsui Pages: 731 - 752 Abstract: We are concerned with stability of numerical methods for delay differential systems of neutral type. In particular, delay-dependent stability of numerical methods is investigated. By means of the H-matrix norm, a necessary and sufficient condition for the asymptotic stability of analytic solution of linear neutral differential systems is derived. Then, based on the argument principle, sufficient conditions for delay-dependent stability of Runge–Kutta and linear multi-step methods are presented, respectively. Furthermore, two algorithms are provided for checking delay-dependent stability of analytical and numerical solutions, respectively. Numerical examples are given to illustrate the main results. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0650-4 Issue No:Vol. 57, No. 3 (2017)

Authors:Ludwig Gauckler; Harry Yserentant Pages: 753 - 770 Abstract: The chemical master equation is a differential equation to model stochastic reaction systems. Its solutions are nonnegative and \(\ell ^1\) -contractive which is inherently related to their interpretation as probability densities. In this note, numerical discretizations of arbitrarily high order are discussed and analyzed that preserve both of these properties simultaneously and without any restriction on the discretization parameters. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0651-3 Issue No:Vol. 57, No. 3 (2017)

Authors:Axel Målqvist; Tony Stillfjord Pages: 787 - 810 Abstract: We consider a system of equations that model the temperature, electric potential and deformation of a thermoviscoelastic body. A typical application is a thermistor; an electrical component that can be used e.g. as a surge protector, temperature sensor or for very precise positioning. We introduce a full discretization based on standard finite elements in space and a semi-implicit Euler-type method in time. For this method we prove optimal convergence orders, i.e. second-order in space and first-order in time. The theoretical results are verified by several numerical experiments in two and three dimensions. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0653-1 Issue No:Vol. 57, No. 3 (2017)

Authors:Christian Mehl; Volker Mehrmann; Punit Sharma Pages: 811 - 843 Abstract: We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0654-0 Issue No:Vol. 57, No. 3 (2017)

Authors:Guangning Tan; Nedialko S. Nedialkov; John D. Pryce Pages: 845 - 865 Abstract: Structural analysis (SA) of a system of differential-algebraic equations (DAEs) is used to determine its index and which equations to be differentiated and how many times. Both Pantelides’s algorithm and Pryce’s \(\varSigma \) -method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates \(\varSigma \) -method’s failures and presents two conversion methods for fixing them. Under certain conditions, both methods reformulate a DAE system on which the \(\varSigma \) -method fails into a locally equivalent problem on which SA is more likely to succeed. Aiming at achieving global equivalence between the original DAE system and the converted one, we provide a rationale for choosing a conversion from the applicable ones. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0655-z Issue No:Vol. 57, No. 3 (2017)

Authors:Georg Muntingh Pages: 867 - 900 Abstract: Pseudo-splines form a family of subdivision schemes that provide a natural blend between interpolating schemes and approximating schemes, including the Dubuc–Deslauriers schemes and B-spline schemes. Using a generating function approach, we derive expressions for the symbols of the symmetric m-ary pseudo-spline subdivision schemes. We show that their masks have positive Fourier transform, making it possible to compute the exact Hölder regularity algebraically as a logarithm of the spectral radius of a matrix. We apply this method to compute the regularity explicitly in some special cases, including the symmetric binary, ternary, and quarternary pseudo-spline schemes. PubDate: 2017-09-01 DOI: 10.1007/s10543-017-0656-y Issue No:Vol. 57, No. 3 (2017)

Authors:Abtin Rahimian; Alex Barnett; Denis Zorin Abstract: We introduce a quadrature scheme—QBKIX —for the ubiquitous high-order accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all distances to the domain boundary as well as on the boundary itself. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no analytic expansion nor addition theorem is required. Moreover, it applies to boundary integrals with singular, weakly singular, and hypersingular kernels. Our work builds upon quadrature by expansion, which approximates the potential by an analytic expansion in the neighborhood of each expansion center. In contrast, we use a sum of fundamental solutions lying on a ring enclosing the neighborhood, and solve a small dense linear system for their coefficients to match the potential on a smaller concentric ring. We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier (elastostatic) kernels in two dimensions (2D) using adaptive, panel-based boundary quadratures on smooth and corner domains. Advantages of the algorithm include its relative simplicity of implementation, immediate extension to new kernels, dimension-independence (allowing simple generalization to 3D), and compatibility with fast algorithms such as the kernel-independent FMM. PubDate: 2017-11-06 DOI: 10.1007/s10543-017-0689-2

Authors:Jeonghun J. Lee Abstract: We propose a new finite element method for a three-field formulation of Biot’s consolidation model in poroelasticity and prove the a priori error estimates. Uniform-in-time error estimates of all the unknowns are obtained for both semidiscrete solutions and fully discrete solutions with the backward Euler time discretization. In contrast to previous results, the implicit constants in our error estimates are uniformly bounded as the Lamé coefficient indicating incompressiblity of poroelastic medium is arbitrarily large, and as the constrained specific storage coefficient is arbitrarily small. Therefore the method does not suffer from the volumetric locking of linear elasticity and provides robust error estimates without additional assumptions on the constrained specific storage coefficient. PubDate: 2017-10-22 DOI: 10.1007/s10543-017-0688-3

Authors:Jana Burkotová; Irena Rachůnková; Ewa B. Weinmüller Abstract: This paper deals with the collocation method applied to solve systems of singular linear ordinary differential equations with variable coefficient matrices and nonsmooth inhomogeneities. The classical stage convergence order is shown to hold for the piecewise polynomial collocation applied to boundary value problems with time singularities of the first kind provided that their solutions are appropriately smooth. The convergence theory is illustrated by numerical examples. PubDate: 2017-10-16 DOI: 10.1007/s10543-017-0686-5

Authors:Pavel Strachota; Michal Beneš Abstract: The Allen–Cahn equation originates in the phase field formulation of phase transition phenomena. It is a reaction-diffusion ODE with a nonlinear reaction term which allows the formation of a diffuse phase interface. We first introduce a model initial boundary-value problem for the isotropic variant of the equation. Its numerical solution by the method of lines is then considered, using a finite volume scheme for spatial discretization. An error estimate is derived for the solution of the resulting semidiscrete scheme. Subsequently, sample numerical simulations in two and three dimensions are presented and the experimental convergence measurement is discussed. PubDate: 2017-10-10 DOI: 10.1007/s10543-017-0687-4

Authors:Tomoya Kemmochi Abstract: In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize the time variable, we use a similar approach to the discrete partial derivative method, which is a structure-preserving method for gradient flows of graphs. For the approximation of curves, we use B-spline curves. Owing to the smoothness of B-spline functions, we can directly address higher order derivatives. Moreover, since B-spline curves require few degrees of freedom, we can reduce the computational cost. In the last part of the paper, we present some numerical examples of the elastic flow, which exhibit topology-changing solutions and more complicated evolution. Videos illustrating our method are available on YouTube. PubDate: 2017-09-27 DOI: 10.1007/s10543-017-0685-6

Authors:Annika Lang; Andreas Petersson; Andreas Thalhammer Abstract: The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler–Maruyama, Milstein, Crank–Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory. PubDate: 2017-09-14 DOI: 10.1007/s10543-017-0684-7

Authors:Tomasz Hrycak; Sebastian Schmutzhard Abstract: This paper studies an approximation to the Chebyshev polynomial \(T_n\) computed via a three-term recurrence in floating-point arithmetic. It is shown that close to either endpoint of the interval \([-1, 1]\) , the numerical approximation coincides with the line tangent to \(T_n\) at that endpoint. From this representation new upper and lower error bounds are derived. PubDate: 2017-09-13 DOI: 10.1007/s10543-017-0683-8

Authors:Jing Gao; Arieh Iserles Abstract: The Filon–Clenshaw–Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency \(\omega \) it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC \(+\) , which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC \(+\) shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages. PubDate: 2017-09-12 DOI: 10.1007/s10543-017-0682-9