A Domain Decomposition Method with Orthogonal Collocation

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Consider the partial differential equation with
and
, subject to the boundary and initial conditions
,
, and
. The solution
is obtained using Mathematica's built-in function NDSolve (solid gray curve) and Chebyshev orthogonal collocation (colored dots). The interval
is divided into three regions identified by the green, red, and blue panes. You can vary the number of Chebyshev collocation points in each region independently as well as the time,
. The behavior of
in the different panes dictates how many collocation points one has to choose from. Indeed, for small times the function
is almost constant (equal to 1) in the red pane region and varies rapidly in the green and blue panes. This indicates that a small number of collocation points is required in the central region while a large number is required near the edges.
Contributed by: Housam Binous (July 2013)
Open content licensed under CC BY-NC-SA
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"A Domain Decomposition Method with Orthogonal Collocation"
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Published: July 8 2013