The concept of pseudospectrum was introduced by Trefethen to explain the behavior of nonnormal operators. Many phenomena (for example, hydrodynamic instability and convergence of iterative methods for linear systems) cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra. The straightforward way to compute pseudospectra involves many applications of the singular value decomposition (SVD). This paper presents several fast continuation methods to calculate pseudospectra of different types of matrices.

In this paper, a gas-kinetic Bhatnagar-Gross-Krook ͑BGK͒ model is constructed for the Rayleigh-Bénard thermal convection in the incompressible flow limit, where the flow field and temperature field are described by two coupled BGK models. Since the collision times in the corresponding BGK models can be different, the Prandtl number can be changed to any value instead of a fixed Prϭ1 in the original BGK model ͓P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 ͑1954͔͒. The two-dimensional Rayleigh-Bénard thermal convection is studied and numerical results are compared with theoretical ones as well as other simulation results. ͓S1063-651X͑99͒00205-6͔

a b s t r a c tSpectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.

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