Authors:Sverre Anmarkrud; Kristian Debrabant; Anne Kværnø Pages: 257 - 280 Abstract: In this paper stochastic partitioned Runge–Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0693-6 Issue No:Vol. 58, No. 2 (2018)

Authors:Dario Fasino; Antonio Fazzi Pages: 281 - 299 Abstract: The Total Least Squares solution of an overdetermined, approximate linear equation \(Ax \approx b\) minimizes a nonlinear function which characterizes the backward error. We devise a variant of the Gauss–Newton iteration with guaranteed convergence to that solution, under classical well-posedness hypotheses. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix A is modified by a rank-one term. In exact arithmetics, the method is equivalent to an inverse power iteration to compute the smallest singular value of the complete matrix \((A\mid b)\) . Geometric and computational properties of the method are analyzed in detail and illustrated by numerical examples. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0678-5 Issue No:Vol. 58, No. 2 (2018)

Authors:Ahmet Guzel; William Layton Pages: 301 - 315 Abstract: This report considers the effect of adding a simple time filter to the fully implicit or backward Euler method. The approach is modular and requires the addition of only one line of additional code. Error estimation and variable time step are straightforward and the individual effect of each step is conceptually clear. The backward Euler method with a curvature reducing time filter induces an equivalent 2-step, second order, A-stable, linear multistep method. PubDate: 2018-06-01 DOI: 10.1007/s10543-018-0695-z Issue No:Vol. 58, No. 2 (2018)

Authors:Tomasz Hrycak; Sebastian Schmutzhard Pages: 317 - 330 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: 2018-06-01 DOI: 10.1007/s10543-017-0683-8 Issue No:Vol. 58, No. 2 (2018)

Authors:Marcel Klinge; Rüdiger Weiner; Helmut Podhaisky Pages: 331 - 345 Abstract: In this paper, explicit peer methods are studied in which some of the stage values are copies of stage values from previous steps. This allows to reduce the number of function calls per step and can be interpreted as being a generalization of the first-same-as-last principle known from Runge–Kutta methods. The variable step size implementation is more complex as the nodes depend on the history of previous step size changes. Optimally zero stable explicit peer methods up to order \(p=8\) are constructed using constraint numerical optimization. In addition the constructed methods are superconvergent of order \(s+1\) for constant step sizes. The new methods show their efficiency in comparison with the Matlab codes ode23, ode45 and ode113 in numerical experiments. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0691-8 Issue No:Vol. 58, No. 2 (2018)

Authors:Jeonghun J. Lee Pages: 347 - 372 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: 2018-06-01 DOI: 10.1007/s10543-017-0688-3 Issue No:Vol. 58, No. 2 (2018)

Authors:Lothar Nannen; Markus Wess Pages: 373 - 395 Abstract: Using perfectly matched layers for the computation of resonances in open systems typically produces artificial or spurious resonances. We analyze the dependency of these artificial resonances with respect to the discretization parameters and the complex scaling function. In particular, we study the differences between a standard frequency independent complex scaling and a frequency dependent one. While the standard scaling leads to a linear eigenvalue problem, the frequency dependent scaling leads to a polynomial one. Our studies show that the location of artificial resonances is more convenient for the frequency dependent scaling than for a standard scaling. Moreover, the artificial resonances of a frequency dependent scaling are less sensitive to the discretization parameters. Hence, the use of a frequency dependent scaling simplifies the choice of the corresponding discretization parameters. PubDate: 2018-06-01 DOI: 10.1007/s10543-018-0694-0 Issue No:Vol. 58, No. 2 (2018)

Authors:Terence J. T. Norton Pages: 397 - 421 Abstract: In choosing a numerical method for the long-time integration of reversible Hamiltonian systems one must take into consideration several key factors: order of the method, ability to preserve invariants of the system, and efficiency of the computation. In this paper, 6th-order composite symmetric general linear methods (COSY-GLMs) are constructed using a generalisation of the composition theory associated with Runge–Kutta methods (RKMs). A novel aspect of this approach involves a nonlinear transformation which is used to convert the GLM to a canonical form in which its starting and finishing methods are trivial. Numerical experiments include efficiency comparisons to symmetric diagonally-implicit RKMs, where it is shown that COSY-GLMs of the same order typically require half the number of function evaluations, as well as long-time computations of both separable and non-separable Hamiltonian systems which demonstrate the preservation properties of the new methods. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0692-7 Issue No:Vol. 58, No. 2 (2018)

Authors:Abtin Rahimian; Alex Barnett; Denis Zorin Pages: 423 - 456 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: 2018-06-01 DOI: 10.1007/s10543-017-0689-2 Issue No:Vol. 58, No. 2 (2018)

Authors:Jeremy Schmitt; Tatiana Shingel; Melvin Leok Pages: 457 - 488 Abstract: In this paper, we present a variational integrator that is based on an approximation of the Euler–Lagrange boundary-value problem via Taylor’s method. This can be viewed as a special case of the shooting-based variational integrator. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0690-9 Issue No:Vol. 58, No. 2 (2018)

Authors:Pavel Strachota; Michal Beneš Pages: 489 - 507 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: 2018-06-01 DOI: 10.1007/s10543-017-0687-4 Issue No:Vol. 58, No. 2 (2018)

Authors:Markus Wahlsten; Jan Nordström Pages: 509 - 529 Abstract: The two dimensional advection–diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry. PubDate: 2018-06-01 DOI: 10.1007/s10543-017-0676-7 Issue No:Vol. 58, No. 2 (2018)

Abstract: The Boris algorithm is a widely used numerical integrator for the motion of particles in a magnetic field. This article proves near-conservation of energy over very long times in the special cases where the magnetic field is constant or the electric potential is quadratic. When none of these assumptions is satisfied, it is illustrated by numerical examples that the numerical energy can have a linear drift or its error can behave like a random walk. If the system has a rotational symmetry and the magnetic field is constant, then also the momentum is approximately preserved over very long times, but in a spatially varying magnetic field this is generally not satisfied. PubDate: 2018-05-30 DOI: 10.1007/s10543-018-0713-1

Authors:Peter Kunkel; Volker Mehrmann Abstract: The solvability and regularity of hybrid differential-algebraic systems (DAEs) is studied, and classical stability estimates are extended to hybrid DAE systems. Different reasons for non-regularity are discussed and appropriate regularization techniques are presented. This includes a generalization of Filippov regularization in the case of so-called chattering. The results are illustrated by several numerical examples. PubDate: 2018-05-24 DOI: 10.1007/s10543-018-0712-2

Authors:Chaobao Huang; Martin Stynes; Na An Abstract: A reaction-diffusion problem with a Caputo time derivative of order \(\alpha \in (0,1)\) is considered. The solution of such a problem has in general a weak singularity near the initial time \(t = 0\) . Some new pointwise bounds on certain derivatives of this solution are derived. The numerical method of the paper uses the well-known L1 discretisation in time on a graded mesh and a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh. Discrete stability of the computed solution is proved. The error analysis is based on a non-trivial projection into the finite element space, which for the first time extends the analysis of the DDG method to non-periodic boundary conditions. The final convergence result implies how an optimal grading of the temporal mesh should be chosen. Numerical results show that our analysis is sharp. PubDate: 2018-05-22 DOI: 10.1007/s10543-018-0707-z

Authors:Gabriel Okša; Yusaku Yamamoto; Martin Bečka; Marián Vajteršic Abstract: The proof of the asymptotic quadratic convergence is provided for the parallel two-sided block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices. The discussion covers the case of well-separated eigenvalues as well as clusters of eigenvalues. Having p processors, each parallel iteration step consists of zeroing 2p off-diagonal blocks chosen by dynamic ordering with the aim to maximize the decrease of the off-diagonal Frobenius norm. Numerical experiments illustrate and confirm the developed theory. PubDate: 2018-05-22 DOI: 10.1007/s10543-018-0711-3

Authors:Hessah Alqahtani; Lothar Reichel Abstract: Multiple orthogonal polynomials generalize standard orthogonal polynomials by requiring orthogonality with respect to several inner products. This paper discusses an application to the approximation of matrix functions and presents quadrature rules that generalize the anti-Gauss rules proposed by Laurie. PubDate: 2018-05-12 DOI: 10.1007/s10543-018-0709-x

Authors:Behnam Hashemi; Yuji Nakatsukasa Abstract: Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun3; a part of Chebfun for numerical computing with 3D functions. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate. PubDate: 2018-05-10 DOI: 10.1007/s10543-018-0710-4

Authors:A. H. Bentbib; M. El Guide; K. Jbilou; E. Onunwor; L. Reichel Abstract: The original version of this article unfortunately contained a mistake. PubDate: 2018-05-04 DOI: 10.1007/s10543-018-0708-y