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In this paper, we consider the convergence criteria for iterative algorithms of Uzawa type for solving linear saddle point problems. Theoretically weaker convergence criteria than before are established for the general case and these are used to deduce conditions for convergence of two special cases: the exact Uzawa algorithm and the linear one-step method. The conclusions given here hold for both symmetric and nonsymmetric saddle point problems. These new sufficient conditions are compared with some known results and illustrated by two examples. Numerical experiments to verify the conclusions in this paper for the preconditioned exact Uzawa algorithm are provided. [Copyright &y& Elsevier]

The development of a novel second-order accurate both in time and space kinetic scheme, called the Kinetic Wave/Particle Split (KWPS) scheme is reported in this paper. It is demonstrated that the first-order KWPS algorithm, while computationally efficient and robust, is very diffusive on a reasonable mesh. The KWPS scheme is derived by employing the following steps: the molecular velocity in the Boltzmann equation is split as a sum of fluid velocity and thermal (peculiar) velocity, thus the flux vector in the Boltzmann equation is divided into a convective part and an acoustic part. The Boltzmann equation is then discretized using an upwind differencing. Moments of the discretized Boltzmann equation with the collision invariant vector and Maxwellian distribution function then yield the KWPS scheme for the Euler equations. The second-order formulation is obtained through the application of a Taylor series expansion of the Maxwellian distribution function to include the first-order derivatives. Numerical test cases, both in 1-D and 2-D, are computed to demonstrate the improved accuracy of the second-order KWPS scheme, while maintaining both the efficiency and robustness of the first-order scheme. It should be noted that the new second-order KWPS formulation cannot be obtained using the standard numerical approach employed in the literature for second-order extension. The methodology employed in this paper can be easily extended to formulate a KWPS scheme to any order of accuracy. [Copyright &y& Elsevier]

Abstract: Central schemes are frequently used for incompressible and compressible flow calculations. The present paper is the first in a forthcoming series where a new approach to a 2nd order accurate Finite Volume scheme operating on Cartesian grids is discussed. Here we start with an adaptively refined Cartesian primal grid in 3D and present a construction technique for the staggered dual grid based on -Voronoi cells. The local refinement constellation on the primal grid leads to a finite number of uniquely defined local patterns on a primal cell. Assembling adjacent local patterns forms the dual grid. All local patterns can be analysed in advance. Later, running the numerical scheme on staggered grids, all necessary geometric information can instantly be retrieved from lookup-tables. The new scheme is compared to established ones in terms of algorithmic complexity and computational effort. [Copyright &y& Elsevier]

Abstract: In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function and partial sums of Gregory''s series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform is sufficiently close to zero, then both approximation methods are accurate. This will contribute for reducing the number of successive square roots of T needed in the inverse scaling and squaring procedure for the matrix logarithm. [Copyright &y& Elsevier]

In this paper, we describe a high-order solver, adaptive in space and time, for the efficient numerical solution of one-dimensional parabolic PDEs. Collocation at Gaussian points is employed for the spatial discretization, using a B-spline basis. A modification of the well-known DAE solver, DASSL is used for the time integration. An a posteriori spatial error estimate is calculated at each successful time step. A new mesh selection strategy based on an equidistribution principle is presented for controlling the spatial error which is balanced with respect to the temporal error. This new mesh adaptation algorithm is shown to be robust, and particularly efficient for problems having solutions with rapid variation. [Copyright &y& Elsevier]