Premium
A Tensor product generalized ADI method for the method of planes
Author(s) -
Dyksen Wayne R.
Publication year - 1988
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690040402
Subject(s) - mathematics , tensor product , hermite polynomials , separable space , elliptic partial differential equation , partial derivative , product (mathematics) , mathematical analysis , partial differential equation , pure mathematics , geometry
We consider solving separable, second order, linear elliptic prtial differential equations in three independent variables. If the partial differential opertor separates into two terms, one depending on x and y, and one depending on z, then we use the method of planes to obtain a discrete problem, which we write in tensor product from as\documentclass{article}\pagestyle{empty}\begin{document}$$ \left({T_z \otimes B_{xy} + I \otimes A_{xy} } \right)C = F $$\end{document}We apply a new interative method, the tensor product generalized alternating direction implicit method, to solve the discrete problem. We study a specific implementation that uses Hermite bicubic collocation in the xy direction and symmetric finite differences in the z direction. We demostrate that this method is a fast and accurate way to solve the large linear systems arising from three‐dimensional elliptic problems.