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Error estimates for Galerkin finite element methods for the Camassa–Holm equation
Springer Science and Business Media LLC - Tập 142 - Trang 833-862 - 2019
We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation
that models one-way propagation of long waves of moderately small amplitude. We
discretize in space the periodic initial-value problem for CH (written in its
original and in system form), using the standard Galerkin finite element method
with smooth splines on a uniform mesh, and prove optimal-order $$L^{2}$$ -error
e...
The numerical evaluation of cauchy principal values of integrals by Romberg integration
Springer Science and Business Media LLC - Tập 21 - Trang 185-192 - 1973
The problem considered is that of evaluating numerically an integral of the form
where the integrand has one or more simple poles in the interval (O,p). Modified
forms of the trapezoidal and mid-ordinate rules, taking account of the
singularities, are obtained; it is then shown that the resulting approximations
can be extrapolated by Romberg's method. Further modifications to deal with the
case wh...
Stability analysis of $\theta$ -methods for neutral functional-differential equations
Springer Science and Business Media LLC - Tập 70 - Trang 473-485 - 1995
This paper deals with the subject of numerical stability for the neutral
functional-differential equation $$ y'(t)=ay(t)+by(qt)+cy'(pt), \qquad t>0. $$
It is proved that numerical solutions generated by $\theta$ -methods are
convergent if $|c|<1$ . However, our numerical experiment suggests that they are
divergent when $|c|$ is large. In order to obtain convergent numerical solutions
when $|c|\geq...
On invariant closed curves for one-step methods
Springer Science and Business Media LLC - Tập 51 - Trang 103-122 - 1987
We show that a one-step method as applied to a dynamical system with a
hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently
small step size. This invariant curve converges to the periodic orbit with the
order of the method and it inherits the stability of the periodic orbit. The
dynamics of the one-step method on the invariant curve can be described by the
rotation number...
An error analysis for radial basis function interpolation
Springer Science and Business Media LLC - Tập 98 - Trang 675-694 - 2004
Radial basis function interpolation refers to a method of interpolation which
writes the interpolant to some given data as a linear combination of the
translates of a single function ϕ and a low degree polynomial. We develop an
error analysis which works well when the Fourier transform of ϕ has a pole of
order 2m at the origin and a zero at ∞ of order 2κ. In case 0≤m≤κ, we derive
error estimates w...
Optimal addition of knots to cubature formulae for planar regions
Springer Science and Business Media LLC - - 1986
On the stability of the incomplete LU-factorizations and characterizations of H-matrices
Springer Science and Business Media LLC - Tập 69 - Trang 321-331 - 1995
Meijerink and van der Vorst [8] have shown that the incomplete LU-factorizations
are numerically stable for M-matrices. Varga, Saff and Mehrmann [16] gave some
characterizations of the H-matrices by using the incomplete LU-factorizations of
them. The purpose of this paper is to show that the incomplete LU-factorizations
of an H-matrix are at least as stable as the complete LU-factorizations of its...
Spline qualocation methods for boundary integral equations
Springer Science and Business Media LLC - Tập 62 - Trang 295-295 - 1992
Construction of basic functions for numerical utilisation of Ritz's method
Springer Science and Business Media LLC - Tập 12 - Trang 435-447 - 1968
Erratum to: Algorithms in unnormalized arithmetic. I. Recurrence relations
Springer Science and Business Media LLC - Tập 7 - Trang 354-354 - 1965
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