It is proved in Benamara-Nikolski [1] that if the spectrum σ($T$) of a contraction$T$with finite defects (rank($I−T$^{*}$T$)=rank ($I−TT$^{*})<∞) does not coincide with $$\bar D$$ , then the contraction is similar to a normal operator if and only if $$C_1 (T) = \mathop {\sup }\limits_{\lambda \in C\backslash \sigma (T)} \parallel (T - \lambda )^{ - 1} \parallel dist(\lambda ,\sigma (T))< \infty .$$

The purpose of this paper is to establish a new method for proving the convergence of the particle method applied to the Camassa-Holm (CH) equation. The CH equation is a strongly nonlinear, bi-Hamiltonian, completely integrable model in the context of shallow water waves. The equation admits solutions that are nonlinear superpositions of traveling waves that have a discontinuity in the first derivative at their peaks and therefore are called peakons. This behavior admits several diverse scientific applications, but introduce difficult numerical challenges. To accurately capture these solutions, one may apply the particle method to the CH equation. Using the concept of space-time bounded variation, we show that the particle solution converges to a global weak solution of the CH equation for positive Radon measure initial data. Read More: http://www.worldscientific.com/doi/abs/10.1142/9789814417099_0033

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.

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We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is <i>O</i>(<i>x</i>^(<i>d</i>+1)) with <i>d</i> the dimension of the physical space. Since there are the same number of mesh points in the <i>x</i><i>t</i> space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only <i>O</i>(1) time. We also extend the algorithm to compute the FTLE