Paul A. Farrell

Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter @e, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates @e-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the @e-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.

Sufficient conditions for the uniform convergence of a difference scheme for a singularly perturbed turning point problem

A number of results exist in the literature for singularly perturbed differential equations without turning points. In particular a number of difference schemes have been proposed that satisfy a stronger than normal convergence criteria known as uniform convergence. This guarantees that the schemes model the boundary layers well. We wish to examine whether these schemes will also be uniformly convergent, if the equation has turning points. To this end we derive sufficient conditions for uniform convergence which are satisfied not only by these schemes but by a more general class of schemes. We show that the rate of convergence is determined by a characteristic parameter of the problem which may be less than one. We confirm these theoretical results by numerical calculations.

A class of singularly perturbed semilinear differential equations with interior layers

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation

Boundary value problems for singularly perturbed semilinear elliptic equations are considered. Special piecewise-uniform meshes are constructed which yield accurate numerical solutions irrespective of the value of the small parameter. Numerical methods composed of standard monotone finite difference operators and these piecewise-uniform meshes are shown theoretically to be uniformly (with respect to the singular perturbation parameter) convergent. Numerical results are also presented, which indicate that in practice the method is first-order accurate.

Continuous and numerical analysis of a multiple boundary turning point problem

A singularly perturbed boundary-value problem with a multiple turning point at a boundary is considered. A representation of the solution is given, and it is used in the construction of a uniform finite-difference scheme. The scheme is a first-order exponentially fitted one. An improved modification on a special discretization mesh is given.

Uniform and optimal schemes for stiff initial-value problems

We formulate a class of difference schemes for stiff initial-value problems, with a small parameter ϵ multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ϵ, that is the solution of the difference scheme uih satisfies |uih−u(xi)| ⪕ Ch, where C is independent of h and ϵ. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |uih−u(xi)| ⪕ C min (h, ϵ). Finally, we show that this class contains higher-order schemes.

PARAMETER-UNIFORM FITTED MESH METHOD FOR QUASILINEAR DIFFERENTIAL EQUATIONS WITH BOUNDARY LAYERS 1

Abstract Singularly perturbed quasilinear boundary value problems exhibiting boundary layers are considered. Special piecewise-uniform meshes are constructed which are fitted to these boundary layers. Numerical methods composed of upwind difference operators and these fitted meshes are shown to be parameter robust, in the sense that the solutions satisfy an error estimate in the maximum norm which is independent of the value of the singular perturbation parameter. Numerical results supporting the theory are presented.

A class of singularly perturbed quasilinear differential equations with interior layers

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular perturbation parameter. Numerical results are presented, which support the theoretical results.

Algorithms for LU decomposition on a shared memory multiprocessor

In this paper we propose an improved algorithm for the parallel LU decomposition of an (m + 1)-banded upper Hessenberg matrix on a shared memory multi-processor, which requires O(2nm2/p) parallel operations, where n is the dimension of the matrix and p is the number of processors. We show that for the special case of tridiagonal matrices this algorithms has a lower operation count than those in the literature and yields the best existing algorithm for the solution of tridiagonal systems of equations.

VIA Communication Performance on a Gigabit Ethernet Cluster

As the technology for high-speedne tworks has evolvedo ver the last decade, the interconnection of commodity computers (e.g., PCs andw orkstations) at gigabit rates has become a reality. However, the improvedp erformance of high-speedne tworks has not been matcheds o far by a proportional improvement in the ability of the TCP/IP protocol stack. As a result the Virtual Interface Architecture (VIA) was developed to remedy this situation by providing a lightweight communication protocol that bypasses operating system interaction, providing low latency and high bandwidth communications for cluster computing. In this paper, we evaluate andc ompare the performance characteristics of both hardware (Giganet) and software (M-VIA) implementations of VIA. In particular, we focus on the performance of the VIA send/receive synchronization mechanism on both uniprocessor andd ual processor systems. The tests were conducted on a Linux cluster of PCs connected by a Gigabit Ethernet network. The performance statistics were collected using a local version of NetPIPE adapted for VIA.