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Summary: This study aimed to estimate the parameters of length‐weight relationship for 16 freshwater fish species that inhabit the Aracá and Demeni Rivers, which are two tributaries of the Negro River (Amazonas state, Brazil). The fishes were sampled using gillnets (15 × 2 m, mesh size 30–120 mm between opposite knots), during 6 days each year, in April 2016, November 2018 and April 2019. The gillnets were placed in the river twice each day; in the morning (5–9 a.m.) and in the evening (5–9 p.m.), with a standardized time of 4 hr. Measurements were taken for standard length (SL – 0.1 cm precision) and total weight (TW – 0.01 g precision). The parameters a and b of the equation WT = a.LTb were estimated. This study also provides new data on of maximum lengths for six species, namely, Argonectes longiceps, Hemiodus atranalis, Chalceus erythrurus, Astrodoras asterifrons, Trachydoras steindachneri and Acarichthys heckelii. [ABSTRACT FROM AUTHOR]

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansions. [ABSTRACT FROM AUTHOR]

Backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations. The numerical solution is formally interpreted as the exact solution of a perturbed differential equation, given as a formal and usually divergent series in powers of the step size. For a rigorous analysis, this series has to be truncated. In this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the long-time behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems. [ABSTRACT FROM AUTHOR]

Asymptotic expansions of the global error of numerical methods are well-understood, if the differential equation is non-stiff. This paper is concerned with such expansions for the implicit Euler method, the linearly implicit Euler method and the linearly implicit mid-point rule, when they are applied to stiff differential equations. In this case perturbation terms are present, whose dominant one is given explicitly. This permits us to better understand the behaviour of extrapolation methods at stiff differential equations. Numerical examples, supporting the theoretical results, are included. [ABSTRACT FROM AUTHOR]

Convergence estimates are given for A(α)-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions. [ABSTRACT FROM AUTHOR]

Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization. [ABSTRACT FROM AUTHOR]

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals by convolution quadrature rules. Here approximations to f* g ( x) on the grid x=0, h, 2h, ..., NhtN h are obtained from a discrete convolution with the values of g on the same grid. The quadrature weights are determined with the help of the Laplace transform of f and a linear multistep method. It is proved that the convolution quadrature method is convergent of the order of the underlying multistep method. [ABSTRACT FROM AUTHOR]

In his fundamental paper on general fixed-stepsize methods, Skeel [6] studied convergence properties, but left the existence of asymptotic expansions as an open problem. In this paper we give a complete answer to this question. For the special cases of one-step and linear multistep methods our proof is shorter than the published ones. Asymptotic expansions are the theoretical base for extrapolation methods. [ABSTRACT FROM AUTHOR]

Discusses the long-time behavior of numerical discretizations of nonlinear parabolic differential equations. Equations of mathematical physics which are governed by a finite-dimensional inertial manifold which attracts solutions at an exponential rate; Runge-Kutta time; Spectral Galerkin space discretizations; Time analyticity; Gevrey regularity of solutions.