#### Title

Analytical Upstream Collocation Solution of a Quadratically Forced Steady-State Convection-Diffusion Equation

#### Document Type

Article

#### Publication Date

6-1-2012

#### DOI

http://dx.doi.org/10.1108/09615531211215747

#### Abstract

*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: extend previously published results (which dealt only with the case of linear forcing) to the case of quadratic forcing; prove the theorem that governs the quadratic case; and then illustrate the results of the theorem using computational examples.

*Findings* – The algorithm developed for the quadratic case dramatically decreases the error (i.e. the difference between the continuous and numerical solutions).

*Research limitations/implications* – Because the methodology successfully extends the linear case to the quadratic case, it is hoped that the method can, indeed, be extended further to more general cases. It is true, however, that the level of complexity rose significantly from the linear case to the quadratic case.

*Practical implications* – Hermite collocation can be used in an optimal way to solve differential equations, especially convection-diffusion equations.

*Originality/value* – Since convection-dominated convection-diffusion equations are difficult to solve numerically, the results in this paper make a valuable contribution to research in this field.

#### Publication Information

Brill, Stephen H. and Smith, Eric. (2012). "Analytical Upstream Collocation Solution of a Quadratically Forced Steady-State Convection-Diffusion Equation". *International Journal of Numerical Methods for Heat & Fluid Flow,** 22*(4), 436-457.