Authors:Chafik Allouch; Driss Sbibih; Mohamed Tahrichi Pages: 3 - 20 Abstract: Abstract Integral equations occur naturally in many fields of mechanics and mathematical physics. In this paper a superconvergent Nyström method has been used for solving one of the most important cases in nonlinear integral equations which is called Urysohn form. Using an interpolatory projection at the set of r Gauss points, it is shown that the proposed method has an order of 3r and one step of iteration improve the convergence order to 4r. The size of the nonlinear system of equations that must be solved to calculate the approximate solution using this method remains the same as the range of the interpolatory projection. Numerical results are given to illustrate the improvement of the order. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0629-6 Issue No:Vol. 57, No. 1 (2017)

Authors:Adam Andersson; Raphael Kruse Pages: 21 - 53 Abstract: Abstract In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be \(\frac{1}{2}\) for the two-step BDF-Maruyama scheme and for the backward Euler–Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the \(\tfrac{3}{2}\) -volatility model from finance and a two dimensional problem related to Galerkin approximation of SPDE, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0624-y Issue No:Vol. 57, No. 1 (2017)

Authors:Winfried Auzinger; Harald Hofstätter; David Ketcheson; Othmar Koch Pages: 55 - 74 Abstract: Abstract We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0626-9 Issue No:Vol. 57, No. 1 (2017)

Authors:Xiao Shan Chen Pages: 75 - 91 Abstract: Abstract The sensitivity of solutions to the periodic Sylvester equation \(A_kX_k-X_{k\oplus 1}B_k\) \(=C_k, k=0, 1, \ldots , K-1,\) is primarily dependent on the quantity introduced by Granat and Kågström (SIAM J Matrix Anal Appl 28:285–300, 2006) in connection with the resolution to periodic invariant subspaces of a product of matrices. In this paper, we give some lower and upper bounds of based on periodic Schur decompositions, which are generalizations of those of the separation between two matrices. Numerical examples are presented to illustrate the theoretical results. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0621-1 Issue No:Vol. 57, No. 1 (2017)

Authors:G. M. Coclite; J. Ridder; N. H. Risebro Pages: 93 - 122 Abstract: Abstract We prove the convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky-Hunter equation on a bounded domain with periodic boundary conditions. The equation models, for example, shallow water waves in a rotating fluid and ultra-short light pulses in optical fibres, and its solutions can develop discontinuities in finite time. Our scheme extends monotone schemes for conservation laws to this equation. The convergence result also provides an existence proof for periodic entropy solutions of the general Ostrovsky-Hunter equation. Uniqueness and an \({\mathscr {O}}({\varDelta x}^{1/2})\) bound on the \(L^1\) error of the numerical solutions are shown using Kružkov’s technique of doubling of variables and a “Kuznetsov type” lemma. Numerical examples confirm the convergence results. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0625-x Issue No:Vol. 57, No. 1 (2017)

Authors:H. de la Cruz; J. C. Jimenez; J. P. Zubelli Pages: 123 - 151 Abstract: Abstract In this work, the performance of Locally Linearized integrators for the numerical simulation of stochastic oscillators driven by random forces is studied. This includes the reproduction of a number of dynamical properties of simple and coupled harmonic oscillators and paths of nonlinear stochastic oscillators in general. Computer experiments illustrate the theoretical findings and advantages of various Locally Linearized schemes in comparison with other conventional integrators. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0620-2 Issue No:Vol. 57, No. 1 (2017)

Authors:Kristian Debrabant; Anne Kværnø Pages: 153 - 168 Abstract: Abstract For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge–Kutta method of order \(p_d\) we obtain methods converging in the mean-square and weak sense with order \(\lfloor p_d/2\rfloor \) . The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0619-8 Issue No:Vol. 57, No. 1 (2017)

Authors:Weiyang Ding; Liqun Qi; Yimin Wei Pages: 169 - 190 Abstract: Abstract In this paper, we show that if a lower-order Hankel tensor is positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS), then its associated higher-order Hankel tensor with the same generating vector, where the higher order is a multiple of the lower order, is also positive semi-definite (or positive definite, or negative semi-definite, or negative definite, or SOS, respectively). Furthermore, in this case, the extremal H-eigenvalues of the higher order tensor are bounded by the extremal H-eigenvalues of the lower order tensor, multiplied with some constants. Based on this inheritance property, we give a concrete sum-of-squares decomposition for each strong Hankel tensor. Then we prove the second inheritance property of Hankel tensors, i.e., a Hankel tensor has no negative (or non-positive, or positive, or nonnegative) H-eigenvalues if the associated Hankel matrix of that Hankel tensor has no negative (or non-positive, or positive, or nonnegative, respectively) eigenvalues. In this case, the extremal H-eigenvalues of the Hankel tensor are also bounded by the extremal eigenvalues of the associated Hankel matrix, multiplied with some constants. The third inheritance property of Hankel tensors is raised as a conjecture. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0622-0 Issue No:Vol. 57, No. 1 (2017)

Authors:Jason Quinn Pages: 207 - 240 Abstract: Abstract A general nonlinear singularly perturbed reaction diffusion differential equation with solutions exhibiting boundary layers is analysed in this paper. The problem is considered as having a stable (attractive) reduced solution that satisfies any one of a comprehensive set of conditions for stable reduced solutions of reaction diffusion problems. A numerical method is presented consisting of a finite difference scheme to be solved over a Shishkin mesh. It is shown that suitable transition points for the Shishkin mesh and the error of the numerical method depend on which stability condition the reduced solution satisfies. Moreover, we show that the error may be affected adversely depending on the stability condition satisfied. Numerical experiments are presented to demonstrate the convergence rate established. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0628-7 Issue No:Vol. 57, No. 1 (2017)

Authors:Xiao Tang; Aiguo Xiao Pages: 241 - 260 Abstract: Abstract In this paper, new weak second-order stochastic Runge–Kutta (SRK) methods for Itô stochastic differential equations (SDEs) with an m-dimensional Wiener process are introduced. Two new explicit SRK methods with weak order 2.0 are proposed. As the main innovation, the new explicit SRK methods have two advantages. First, only three evaluations of each diffusion coefficient are needed in per step. Second, the number of necessary random variables which have to be simulated is only \(m+2\) for each step. Compared to well-known explicit SRK methods, these good properties can be used to reduce the computational effort. Our methods are compared with other well-known explicit weak second-order SRK methods in numerical experiments. And the numerical results show that the computational efficiency of our methods is better than other methods. PubDate: 2017-03-01 DOI: 10.1007/s10543-016-0618-9 Issue No:Vol. 57, No. 1 (2017)

Authors:Georg Muntingh Abstract: 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: 2017-04-13 DOI: 10.1007/s10543-017-0656-y

Authors:Guangning Tan; Nedialko S. Nedialkov; John D. Pryce Abstract: 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: 2017-04-07 DOI: 10.1007/s10543-017-0655-z

Authors:Axel Målqvist; Tony Stillfjord Abstract: 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: 2017-03-15 DOI: 10.1007/s10543-017-0653-1

Authors:Christian Mehl; Volker Mehrmann; Punit Sharma Abstract: 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: 2017-03-14 DOI: 10.1007/s10543-017-0654-0

Authors:Ludwig Gauckler; Harry Yserentant Abstract: 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: 2017-03-10 DOI: 10.1007/s10543-017-0651-3

Authors:Guang-Da Hu; Taketomo Mitsui Abstract: 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: 2017-03-06 DOI: 10.1007/s10543-017-0650-4

Authors:Xingjun Luo; Chunmei Zeng; Suhang Yang; Rong Zhang Abstract: 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: 2017-03-06 DOI: 10.1007/s10543-017-0649-x

Authors:Sebastian Franz Abstract: Abstract We present a minor modification of of the stabilisation term in the local projection stabilisation (LPS) method for stabilising singularly perturbed problems on layer-adapted meshes, such that the error can be estimated uniformly and in optimal order in the associated LPS-norm, which is stronger than the standard energy norm. Numerical results confirm the theoretical findings. PubDate: 2017-03-01 DOI: 10.1007/s10543-017-0652-2