Authors:Mohammad Asadzadeh; Christoffer Standar Pages: 5 - 26 Abstract: This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic Vlasov–Maxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0666-9 Issue No:Vol. 58, No. 1 (2018)

Authors:M. Irene Falcão; Fernando Miranda; Ricardo Severino; M. Joana Soares Pages: 51 - 72 Abstract: In this paper we focus on computational aspects associated with polynomial problems in the ring of one-sided quaternionic polynomials. The complexity and error bounds of quaternion arithmetic are considered and several evaluation schemes are analyzed from their complexity point of view. The numerical stability of generalized Horner’s and Goertzel’s algorithms to evaluate polynomials with quaternion floating-point coefficients is addressed. Numerical tests illustrate the behavior of the algorithms from the point of view of performance and accuracy. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0667-8 Issue No:Vol. 58, No. 1 (2018)

Authors:Liviu I. Ignat; Alejandro Pozo Pages: 73 - 102 Abstract: In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0673-x Issue No:Vol. 58, No. 1 (2018)

Authors:Rakesh Kumar Pages: 103 - 132 Abstract: In this article, we have proposed a septic B-spline quasi-interpolation (SeBSQI) based numerical scheme for the modified Burgers’ equation. The SeBSQI scheme maintains eighth order accuracy for the smooth solution, but fails to maintain a non-oscillatory profile when the solution has discontinuities or sharp variations. To ensure the non-oscillatory profile of the solution, we have proposed an adaptive SeBSQI (ASeBSQI) scheme for the modified Burgers’ equation. The ASeBSQI scheme maintains higher order accuracy in the smooth regions using SeBSQI approximation and in regions with discontinuities or sharp variations, 5th order weighted essentially non-oscillatory (WENO) reconstruction is used to preserve a non-oscillatory profile. To identify discontinuous or sharp variation regions, a weak local truncation error based smooth indicator is proposed for the modified Burgers’ equation. For the temporal derivative, we have considered the Runge–Kutta method of order four. We have shown numerically that the ASeBSQI scheme preserves the convergence rate of the SeBSQI and it converges to the exact solution with convergence rate eight. We have performed numerical experiments to validate the proposed scheme. The numerical experiments demonstrate an improvement in accuracy and efficiency of the proposed schemes over the WENO5 and septic B-spline collocation schemes. The ASeBSQI scheme is also tested for one-dimensional Euler equations. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0675-8 Issue No:Vol. 58, No. 1 (2018)

Authors:Giampaolo Mele; Elias Jarlebring Pages: 133 - 162 Abstract: An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0671-z Issue No:Vol. 58, No. 1 (2018)

Authors:Xinyan Niu; Jianbo Cui; Jialin Hong; Zhihui Liu Pages: 163 - 178 Abstract: We construct stochastic pseudo-symplectic methods and analyze their pseudo-symplectic orders for stochastic Hamiltonian systems with additive noises in this paper. All of these methods are explicit so that the numerical implementations become much easier than implicit methods. Through the numerical experiments, we find that these methods have desired properties in accuracy and stability as well as the preservation of the symplectic structure of the systems. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0668-7 Issue No:Vol. 58, No. 1 (2018)

Authors:Sotirios E. Notaris Pages: 179 - 198 Abstract: It is well known that the Gauss–Kronrod quadrature formula does not always exist with real and distinct nodes and positive weights. In 1996, in an attempt to find an alternative to the Gauss–Kronrod formula for estimating the error of the Gauss quadrature formula, Laurie constructed the anti-Gaussian quadrature formula, which always has real and distinct nodes and positive weights. First, we give a description and prove the most important properties of the anti-Gaussian formula, by applying a different approach than that of Laurie. Then, we consider a measure such that the respective (monic) orthogonal polynomials, above a specific index, satisfy a three-term recurrence relation with constant coefficients. We show that for a measure of this kind the nodes of the anti-Gaussian formula are the zeros of the respective Stieltjes polynomial, while the resulting averaged Gaussian quadrature formula is precisely the corresponding Gauss–Kronrod formula, having elevated degree of exactness. Moreover, we show, by a new method, that a symmetric Gauss–Lobatto quadrature formula is a modified anti-Gaussian formula, and we specialize our results to the measures with constant recurrence coefficients. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0672-y Issue No:Vol. 58, No. 1 (2018)

Authors:Tomoaki Okayama Pages: 199 - 220 Abstract: A Sinc-collocation method was proposed by Stenger, who also gave a theoretical analysis of the method in the case of a “scalar” equation. This paper extends the theoretical results to the case of a “system” of equations. Furthermore, this paper proposes a more efficient method by replacing the variable transformation employed in Stenger’s method. The efficiency was confirmed by both a theoretical analysis and some numerical experiments. In addition to the existing and newly proposed Sinc-collocation methods, this paper also gives similar theoretical results for the Sinc-Nyström methods proposed by Nurmuhammad et al. In terms of computational cost, the newly proposed Sinc-collocation method is the most efficient among these methods. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0663-z Issue No:Vol. 58, No. 1 (2018)

Authors:Jin Zhang; Xiaowei Liu Pages: 221 - 246 Abstract: In this paper, we present a pointwise convergence analysis for a streamline diffusion finite element method (SDFEM) on a Shishkin triangular mesh. We prove that the method is uniformly convergent with a pointwise accuracy of order almost 7/4 (up to a logarithmic factor) away from the subdomain where the layers intersect. Finally, numerical experiments support our theoretical results. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0661-1 Issue No:Vol. 58, No. 1 (2018)

Authors:Nils Henrik Risebro; Christoph Schwab; Franziska Weber Pages: 247 - 255 Abstract: An error in [4, Theorem 4.1, 4.5, Corollary 4.5] is corrected. There, in the Monte Carlo error bounds for front tracking for scalar conservation laws with random input data, 2-integrability in a Banach space of type 1 was assumed. In providing the corrected convergence rate bounds and error versus work analysis of multilevel Monte Carlo front-tracking methods, we also generalize [4] to q-integrability of the random entropy solution for some \(1<q\le 2\) , allowing possibly infinite variance of the random entropy solutions of the scalar conservation law. PubDate: 2018-03-01 DOI: 10.1007/s10543-017-0670-0 Issue No:Vol. 58, No. 1 (2018)

Authors:Andrew J. Steyer; Erik S. Van Vleck Abstract: Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results. PubDate: 2018-04-13 DOI: 10.1007/s10543-018-0704-2

Authors:A. H. Bentbib; M. El Guide; K. Jbilou; E. Onunwor; L. Reichel Abstract: This work discusses four algorithms for the solution of linear discrete ill-posed problems with several right-hand side vectors. These algorithms can be applied, for instance, to multi-channel image restoration when the image degradation model is described by a linear system of equations with multiple right-hand sides that are contaminated by errors. Two of the algorithms are block generalizations of the standard Golub–Kahan bidiagonalization method with the block size equal to the number of channels. One algorithm uses standard Golub–Kahan bidiagonalization without restarts for all right-hand sides. These schemes are compared to standard Golub–Kahan bidiagonalization applied to each right-hand side independently. Tikhonov regularization is used to avoid severe error propagation. Numerical examples illustrate the performance of these algorithms. Applications include the restoration of color images. PubDate: 2018-04-12 DOI: 10.1007/s10543-018-0706-0

Authors:Stefan Kopecz; Andreas Meister Abstract: Modified Patankar–Runge–Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production–destruction systems. They adapt explicit Runge–Kutta schemes to ensure positivity and conservation irrespective of the time step size. The first two members of this class, the first order MPE scheme and the second order MPRK22(1) scheme, have been successfully applied in a large number of applications. Recently, a general definition of MPRK schemes was introduced and necessary as well as sufficient conditions to obtain first and second order MPRK schemes were presented. In this paper we derive necessary and sufficient conditions for third order MPRK schemes and introduce the first family of such schemes. The theoretical results are confirmed by numerical experiments considering linear and nonlinear as well as nonstiff and stiff systems of differential equations. PubDate: 2018-03-27 DOI: 10.1007/s10543-018-0705-1

Authors:Allal Guessab; Boris Semisalov Abstract: In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in \(\mathbb {R}^d\) are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results. PubDate: 2018-03-24 DOI: 10.1007/s10543-018-0703-3

Authors:Shuai Zhu; Shilie Weng Abstract: This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenience. The standard SDC method is used in parallel with a rough initial guess and a Picard integral equation with high precision initial condition is acted as a compensator. The goal of this paper is to show how these processes can be parallelized and how to improve the efficiency. During the SDC step an implicit or semi-implicit method can be used for stiff problems which is always time-consuming, therefore that’s why we do this procedure in parallel. Due to a better initial condition of parallel intervals after the SDC step, the goal of compensation step is to get a better approximation and also avoid of solving an implicit problem again. During the compensation step an explicit Picard scheme is taken based on the numerical integration with polynomial interpolation on Gauss Radau II nodes, which is almost no time consumption, obviously, that’s why we do this procedure in serial. The convergency analysis and the parallel efficiency of our method are also discussed. Several numerical experiments and an application for simulation Allen–Cahn equation are presented to show the accuracy, stability, convergence order and efficiency features of para-SDC method. PubDate: 2018-03-14 DOI: 10.1007/s10543-018-0702-4

Authors:Yanmei Chen; Xiaofei Peng; Wen Li Abstract: In this paper, we present some uniform relative perturbation bounds for eigenvalues and eigenspaces of diagonalizable matrices under additive and multiplicative perturbations. Some existing perturbation bounds can be improved based on the new bounds. Numerical experiments are given to demonstrate the advantage of the new bounds. PubDate: 2018-03-08 DOI: 10.1007/s10543-018-0701-5

Authors:Xue-Lei Lin; Michael K. Ng; Hai-Wei Sun Abstract: In this paper, we propose an efficient preconditioner for the linear systems arising from the one-sided space fractional diffusion equation with variable coefficients. The shifted Gr \(\ddot{\mathrm{u}}\) nwald formula is employed to discretize the one-sided Riemann–Liouville fractional derivative. The matrix structure of resulting linear systems is Toeplitz-like, which is a summation of an identity matrix and a diagonal-times-nonsymmetric-Toeplitz matrix. A diagonal-times-nonsymmetric-Toeplitz preconditioner is proposed to reduce the condition number of the Toeplitz-like matrix, where the diagonal part comes from the variable coefficients and the nonsymmetric Toeplitz part comes from the Riemann–Liouville derivative. Theoretically, we show that the condition number of the preconditioned matrix is uniformly bounded by a constant independent of discretization step-sizes under certain assumptions on the coefficient function. Due to the uniformly bounded condition number, the Krylov subspace method for the preconditioned linear systems converges linearly and independently on discretization step-sizes. Numerical results are reported to show the efficiency of the proposed preconditioner and to demonstrate its superiority over other tested preconditioners. PubDate: 2018-02-27 DOI: 10.1007/s10543-018-0699-8

Authors:Pengde Wang; Chengming Huang Abstract: This paper proposes and analyzes a high-order implicit-explicit difference scheme for the nonlinear complex fractional Ginzburg–Landau equation involving the Riesz fractional derivative. For the time discretization, the second-order backward differentiation formula combined with the explicit second-order Gear’s extrapolation is adopted. While for the space discretization, a fourth-order fractional quasi-compact method is used to approximate the Riesz fractional derivative. The scheme is efficient in the sense that, at each time step, only a linear system with a coefficient matrix independent of the time level needs to be solved. Despite of the explicit treatment of the nonlinear term, the scheme is shown to be unconditionally convergent in the \(l^2_h\) norm, semi- \(H^{\alpha /2}_h\) norm and \(l^\infty _h\) norm at the order of \(O(\tau ^2+h^4)\) with \(\tau \) time step and h mesh size. Numerical tests are provided to confirm the accuracy and efficiency of the scheme. PubDate: 2018-02-14 DOI: 10.1007/s10543-018-0698-9

Authors:J. M. Carnicer; C. Godés Abstract: The Lebesgue constant is a measure for the stability of the Lagrange interpolation. The decomposition of the Lagrange interpolation operator in their even and odd parts with respect to the last variable can be used to find a relation between the Lebesgue constant for a space of polynomials and the corresponding Lebesgue constants for subspaces of even and odd polynomials. It is shown that such a decomposition preserves the stability properties of the Lagrange interpolation operator. We use the Lebesgue functions to provide pointwise quantitative measures of the stability properties and illustrate with examples the behaviour in simple cases. PubDate: 2017-12-01 DOI: 10.1007/s10543-017-0674-9