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Full-Text Articles in Physical Sciences and Mathematics
Analytical Upstream Collocation Solution Of A Quadratically Forced Steady-State Convection-Diffusion Equation, Stephen Brill, Eric Smith
Analytical Upstream Collocation Solution Of A Quadratically Forced Steady-State Convection-Diffusion Equation, Stephen Brill, Eric Smith
Stephen H. Brill
Purpose – The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady-state convection-diffusion equation in one spatial dimension with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method “upstream weighting” of the convective term is used in an optimal way. The authors also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term. Design/methodology/approach – The authors: …
Eigenvalue Analysis Of A Block Red-Black Gauss–Seidel Preconditioner Applied To The Hermite Collocation Discretization Of Poisson's Equation, Stephen Brill, George Pinder
Eigenvalue Analysis Of A Block Red-Black Gauss–Seidel Preconditioner Applied To The Hermite Collocation Discretization Of Poisson's Equation, Stephen Brill, George Pinder
Stephen H. Brill
This article is concerned with the numerical solution of Poisson's equation with Dirichlet boundary conditions, defined on the unit square, discretized by Hermite collocation with uniform mesh. In [1], it was demonstrated that the Bi-CGSTAB method of van der Vorst [2] with block Red-Black Gauss–Seidel (RBGS) preconditioner is an efficient method to solve this problem. In this article, we derive analytic formulae for the eigenvalues that control the rate at which the Bi-CGSTAB/RBGS method converges. These formulae, which depend upon the location of the collocation points, can be utilized to determine where the collocation points should be placed in order …
Optimal Hermite Collocation Applied To A One-Dimensional Convection-Diffusion Equation Using An Adaptive Hybrid Optimization Algorithm, Karen Ricciardi, Stephen Brill
Optimal Hermite Collocation Applied To A One-Dimensional Convection-Diffusion Equation Using An Adaptive Hybrid Optimization Algorithm, Karen Ricciardi, Stephen Brill
Stephen H. Brill
Purpose – The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady-state one-dimensional convection-diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite-element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem …
Analysis Of A Block Red-Black Preconditioner Applied To The Hermite Collocation Discretization Of A Model Parabolic Equation, Stephen Brill, George Pinder
Analysis Of A Block Red-Black Preconditioner Applied To The Hermite Collocation Discretization Of A Model Parabolic Equation, Stephen Brill, George Pinder
Stephen H. Brill
We are concerned with the numerical solution of a model parabolic partial differential equation (PDE) in two spatial dimensions, discretized by Hermite collocation. In order to efficiently solve the resulting systems of linear algebraic equations, we choose the Bi-CGSTAB method of van der Vorst (1992) with block Red-Black Gauss-Seidel (RBGS) preconditioner. In this article, we give analytic formulae for the eigenvalues that control the rate at which Bi-CGSTAB/RBGS converges. These formulae, which depend on the location of the collocation points, can be utilized to determine where the collocation points should be placed in order to make the Bi-CGSTAB/RBGS method converge …