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1078-0947

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## Discrete & Continuous Dynamical Systems - A

March 2014 , Volume 34 , Issue 3

Special issue dedicated to Arieh Iserles on the occasion of his 65th birthday

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2014, 34(3): i-ii
doi: 10.3934/dcds.2014.34.3i

*+*[Abstract](1237)*+*[PDF](95.7KB)**Abstract:**

Arieh Iserles was born in Poland, on September 2, 1947. He was educated in Israel, where he received BSc and MSc degrees from the Hebrew University and obtained his PhD degree under the supervision of Giacomo Della Riccia at Ben Gurion University with the dissertation

*Numerical Solution of Stiff Differential Equations*(1978). He first arrived in Cambridge, in 1978 and has remained there ever since. He has successively been Junior and Senior Research Fellow at King's College, and Lecturer (1987) and Professor (1999) at Cambridge University where he holds a chair in Numerical Analysis and Differential Equations. Arieh has received many honours, in particuluar the Lars Onsager Medal (1999) from the Nowegian University of Science and Technology and the David G. Crighton Medal (2012) from the London Mathematical Society and the Institute of Mathematics and its Applications. He holds Honorary Professorships at Huazhong University of Science and Technology, Wuhan, since 2002 and Jilin University, Changchun, since 2004.

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2014, 34(3): 883-901
doi: 10.3934/dcds.2014.34.883

*+*[Abstract](1628)*+*[PDF](1178.7KB)**Abstract:**

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{i\omega x}$ on the interval $[-1,1]$. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency $\omega$. However, accuracy is maintained for all values of $\omega$ and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as $\omega \to 0$. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.

2014, 34(3): 903-914
doi: 10.3934/dcds.2014.34.903

*+*[Abstract](1755)*+*[PDF](517.8KB)**Abstract:**

We present an algorithm based on Automatic Differentiation for computing general B-series of vector fields $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^n$. The algorithm has a computational complexity depending linearly on $n$, and provides a practical way of computing B-series up to a moderately high order $d$. Compared to Automatic Differentiation for computing Taylor series solutions of differential equations, the proposed algorithm is more general, since it can compute

*any*B-series. However the computational cost of the proposed algorithm grows much faster in $d$ than a Taylor series method, thus very high order B-series are not tractable by this approach.

2014, 34(3): 915-929
doi: 10.3934/dcds.2014.34.915

*+*[Abstract](2086)*+*[PDF](343.3KB)**Abstract:**

We study the high-oscillation properties of solutions to integral equations associated with two classes of Volterra integral operators: compact operators with highly oscillatory kernels that are either smooth or weakly singular, and noncompact cordial Volterra integral operators with highly oscillatory kernels. In the latter case the focus is on the dependence of the (uncountable) spectrum on the oscillation parameter. It is shown that the results derived in this paper merely open a window to a general theory of solutions of highly oscillatory Volterra integral equations, and many questions remain to be answered.

2014, 34(3): 931-957
doi: 10.3934/dcds.2014.34.931

*+*[Abstract](2330)*+*[PDF](1772.8KB)**Abstract:**

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the $H^{-1}$-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.

2014, 34(3): 959-975
doi: 10.3934/dcds.2014.34.959

*+*[Abstract](1702)*+*[PDF](474.9KB)**Abstract:**

A perturbative procedure based on the Lie--Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear differential equations with periodic coefficients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous system and also to its characteristic exponents. The procedure is computationally well adapted and converges for sufficiently small values of the perturbation parameter. Moreover, when the system evolves in a Lie group, the approximations also belong to the same Lie group, thus preserving qualitative properties of the exact solution.

2014, 34(3): 977-990
doi: 10.3934/dcds.2014.34.977

*+*[Abstract](1979)*+*[PDF](1492.0KB)**Abstract:**

The discrete gradient approach is generalized to yield first integral preserving methods for differential equations in Lie groups.

2014, 34(3): 991-1008
doi: 10.3934/dcds.2014.34.991

*+*[Abstract](1997)*+*[PDF](1845.2KB)**Abstract:**

When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon whether the time is rational or irrational relative to the length of the interval, thus producing the Talbot effect of dispersive quantization and fractalization. The goal here is to show that these remarkable phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations, based on operator splitting methods, of the nonlinear Schrödinger and Korteweg--deVries equations with step function initial data and periodic boundary conditions. For the integrable nonlinear Schrödinger equation, our observations have been rigorously confirmed in a recent paper of Erdoǧan and Tzirakis, [10].

2014, 34(3): 1009-1020
doi: 10.3934/dcds.2014.34.1009

*+*[Abstract](1935)*+*[PDF](371.9KB)**Abstract:**

We propose a model for swarming (i.e., cohesion preserving) that shares all the good properties of the CS-model for flocking. In particular, we show for this model that under strong interactions of the agents swarming unconditionally occurs and that, furthermore, it does so in a collision avoiding manner. We also show that under weak interactions the same holds true provided the initial state of the population (their positions and velocities) satisfies some explicit inequalities.

2014, 34(3): 1021-1040
doi: 10.3934/dcds.2014.34.1021

*+*[Abstract](1530)*+*[PDF](311.6KB)**Abstract:**

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a linear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik--Plebański--Strichartz formula for this logarithm.

2014, 34(3): 1041-1060
doi: 10.3934/dcds.2014.34.1041

*+*[Abstract](1484)*+*[PDF](477.9KB)**Abstract:**

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive sample-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric.

In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and show that these operators approximate continuous SVFs. The rate of approximation of Hölder continuous SVFs by the adapted Bernstein operators is studied and shown to be asymptotically equal to the one for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.

2014, 34(3): 1061-1078
doi: 10.3934/dcds.2014.34.1061

*+*[Abstract](1685)*+*[PDF](2715.3KB)**Abstract:**

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and

*ad hoc*filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these

*ad hoc*filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.

2014, 34(3): 1079-1097
doi: 10.3934/dcds.2014.34.1079

*+*[Abstract](2485)*+*[PDF](771.0KB)**Abstract:**

We present a new numerical multiscale integrator for stiff and highly oscillatory dynamical systems. The new algorithm can be seen as an improved version of the seamless Heterogeneous Multiscale Method by E, Ren, and Vanden-Eijnden and the method FLAVORS by Tao, Owhadi, and Marsden. It approximates slowly changing quantities in the solution with higher accuracy than these other methods while maintaining the same computational complexity. To achieve higher accuracy, it uses variable mesoscopic time steps which are determined by a special function satisfying moment and regularity conditions. Detailed analytical and numerical comparison between the different methods are given.

2014, 34(3): 1099-1104
doi: 10.3934/dcds.2014.34.1099

*+*[Abstract](1927)*+*[PDF](265.1KB)**Abstract:**

We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.

2014, 34(3): 1105-1120
doi: 10.3934/dcds.2014.34.1105

*+*[Abstract](1710)*+*[PDF](580.1KB)**Abstract:**

We revisit an algorithm by Skeel

*et al.*[5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory.

2014, 34(3): 1121-1130
doi: 10.3934/dcds.2014.34.1121

*+*[Abstract](1482)*+*[PDF](492.2KB)**Abstract:**

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. The family includes the

*contact oscillator*, which has been used as a test problem for numerical methods for nonholonomic mechanics. The systems under study constitute simple models for continuously variable transmission gearboxes.

The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. By using reversible Kolmogorov--Arnold--Moser theory, we then establish preservation of invariant tori for reversible perturbations. This result explains previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.

2014, 34(3): 1131-1146
doi: 10.3934/dcds.2014.34.1131

*+*[Abstract](1664)*+*[PDF](2413.7KB)**Abstract:**

High order variants of the classical Størmer-Cowell methods are still a popular class of methods for computations in celestial mechanics. In this work we shall investigate the absolute stability of Størmer-Cowell methods close to zero, and present a characterization of the stability of methods of all orders. In particular, we show that many methods are not absolutely stable at any point in a neighborhood of the origin.

2014, 34(3): 1147-1170
doi: 10.3934/dcds.2014.34.1147

*+*[Abstract](2049)*+*[PDF](859.8KB)**Abstract:**

In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large class of discrete gradient methods that the numerical solution exists and is locally unique, and that for arbitrary $p\in \mathbb{N}$ we may construct a method that is of order $p$. In the proofs of these results we also show that the constants in the time step constraint and the error bounds may be chosen independently from the distance to critical points of the first integral.

In the case when the first integral is quadratic, for arbitrary $p \in \mathbb{N}$, we have devised a new method that is linearly implicit at each time step and of order $p$. A numerical example suggests that this new method has advantages in terms of efficiency.

2014, 34(3): 1171-1182
doi: 10.3934/dcds.2014.34.1171

*+*[Abstract](1451)*+*[PDF](331.5KB)**Abstract:**

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the $2$-sphere, we show that for each $n$, $f$ has at least $2^{n}$ periodic points of period $n$.

2014, 34(3): 1183-1210
doi: 10.3934/dcds.2014.34.1183

*+*[Abstract](1594)*+*[PDF](516.0KB)**Abstract:**

We consider the Landau--Kolmogorov problem on a finite interval which is to find an exact bound for $\|f^{(k)}\|$, for $0 < k < n$, given bounds $\|f\| \le 1$ and $\|f^{(n)}\| \le \sigma$, with $\|\cdot\|$ being the max-norm on $[-1,1]$. In 1972, Karlin conjectured that this bound is attained at the end-point of the interval by a certain Zolotarev polynomial or spline, and that was proved for a number of particular values of $n$ or $\sigma$. Here, we provide a complete proof of this conjecture in the polynomial case, i.e. for $0 \le \sigma \le \sigma_n := \|T_n^{(n)}\|$, where $T_n$ is the Chebyshev polynomial of degree $n$. In addition, we prove a certain Schur-type estimate which is of independent interest.

2014, 34(3): 1211-1228
doi: 10.3934/dcds.2014.34.1211

*+*[Abstract](1935)*+*[PDF](755.4KB)**Abstract:**

Symplectic integration of stochastic Hamiltonian systems is a developing branch of stochastic numerical analysis. In the present paper, a stochastic generating function approach is proposed, based on the derivation of stochastic Hamilton-Jacobi PDEs satisfied by the generating functions, and a method of approximating solutions to them. Thus, a systematic approach of constructing stochastic symplectic methods is provided. As validation, numerical tests on several stochastic Hamiltonian systems are performed, where some symplectic schemes are constructed via stochastic generating functions. Moreover, generating functions for some known stochastic symplectic mappings are given.

2014, 34(3): 1229-1249
doi: 10.3934/dcds.2014.34.1229

*+*[Abstract](1862)*+*[PDF](418.2KB)**Abstract:**

In this paper, we study generating forms and generating functions for volume preserving mappings in $\mathbf{R}^n$. We derive some parametric classes of volume preserving numerical schemes for divergence free vector fields. In passing, by extension of the Poincaré generating function and a change of variables, we obtained symplectic equivalent of the theta-method for differential equations, which includes the implicit midpoint rule and symplectic Euler A and B methods as special cases.

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