A Domain Decomposition Method with Orthogonal Collocation

collocation points

first interval

29

second interval

39

third interval

23

time

0.17

Consider the partial differential equation

∂u

∂t

=

2

∂

u

∂

2

x

with

t≥0

and

-2≤x≤2

, subject to the boundary and initial conditions

u(x=2,t)=0

,

u(x=-2,t)=0

, and

u(x,t=0)=1

. The solution

u(x,t)

is obtained using Mathematica's built-in function NDSolve (solid gray curve) and Chebyshev orthogonal collocation (colored dots). The interval

[-2,2]

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,

t

. The behavior of

u(x,t)

in the different panes dictates how many collocation points one has to choose from. Indeed, for small times the function

u(x,t)

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.