Examples shown that IIM produces second order accurate
solution and the first order derivatives in 1D
Nnenna Anoruo, Kevin Luna, William Reese, Justin Rivera
Faculty Mentor: Dr.Zhilin Li
NC State University REU 2015 (June 1-July 31)
1
Introduction
Here are some sample computations demonstrating the second order accurate derivative at the
interface for the 1-D interface problem presented in ”The Immersed Interface Method” by Dr.Li
by using a two sided method directly related to the IIM. The 1-D Interface problem where x ∈
(0, α) ∪ (α, 1) is often expressed as:
(βx u)x − σu = f
In the examples presented here, all the functions have finite jumps at the interface α. The same β
and σ are used throughout, and the proper Dirichlet boundary condition according the the actual
solution are always chosen. Note

1 + cos(x) if x ≤ α
2
β(x) =

1 + x2 if x > α
and
σ(x) =
ln(2 + x)
.
2

(1 + x)2
if x ≤ α
if
x>α
Example 1
Consider the problem where

(120)2 (−( cos(x) + 1)) sin(120x) − 120 sin(x) cos(120x) − (1 + x)2 cos(120x) if x < α
2
2
f (x) =

2
(30)(−30(x + 1) cos(30x) − 2x sin(30x)) − ln(2 + x) cos(30x) if x < α.
The exact solution can be shown to be
u(x) =

sin(120x) if x ≤ α
cos(30x) if
1
x>α
(β ux (x)) x -σ(x)u(x)=f(x) + cδ (x-α) Approximation and Exact Solution with DBC
1.5
1
×10
-3
Error (difference) Plot
Exact
Approximation
0
1
-1
-2
u(x)
0.5
-3
0
-4
-5
-0.5
-6
-1
-7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Figure 1: (a): An example of a computed solution with n = 1280 grid divisions. Note the actual
solution has its discontinuity connected by the plotter. (b): The error difference of the solution and
actual solution at the grid points.
loglog Plot of U +x (α) Error
Solution Error Plot on a log-log Scale for the IIM
1
1
0
0
log10( ||U +x (α)-u +x (α)|| ∞ )
log10( ||U-u||∞ )
-1
-2
-3
-4
-1
-2
-3
-4
-5
-5
Two-Sided Method(IIM)
One-sided method
Immersed Interface Method
-5
-4
-3
-2
-1
0
1
-5
log10(h)
-4
-3
-2
-1
0
1
log10(h)
Figure 2: (a): Grid refinement analysis of the approximate solution with 15 refinements starting
at n = 20. Note that the axis has been adjusted so some points may not be visible. The slope of
the regression line fit is 2.1 (b):Grid refinement analysis of the right derivative at the interface with
the one-sided and two-sided(IIM) methods. Note that a linear regression fit shows that the slope
of the line of best fit is 1.08 and 1.90 for the one-sided and two-sided methods respectively. This
shows the IIM two sided method gave an order 2 accurate approximation while standard one-sided
methods gave a first order accurate approximation.
3
Example 2
Consider the problem where

cos(x)
120

(120)2 (−( 2 + 1)) sin(120x) − 2 sin(x) cos(120x) − (1 + x)2 sin(120x) if x ≤ α
f (x) =
2

 x +200x−1 − ln(2 + x) ln(100 + x) if x > α.
(x+100)2
2
The exact solution can be shown to be
u(x) =

sin(120x) if
ln(100 + x)
(β ux (x)) x -σ(x)u(x)=f(x) + cδ (x-α) Approximation and Exact Solution with DBC
5
8
x≤α
if
x>α
×10 -3
Error (difference) Plot
Exact
Approximation
7
4
6
3
5
u(x)
U-u(x)
2
4
1
3
0
2
-1
1
-2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
x
Figure 3: (a): An example of a computed solution with n = 1280 grid divisions. Note the actual
solution has its discontinuity connected by the plotter. Note that the axis has been adjusted so
some points may not be visible. (b): The error difference of the solution and actual solution at the
grid points.
loglog Plot of U -x (α) Error
Solution Error Plot on a log-log Scale for the IIM
1
0
-0.5
0
-1
log10( ||U -x(α)-u -x(α)|| ∞ )
log10( ||U-u||∞ )
-1
-2
-3
-1.5
-2
-2.5
-3
-3.5
-4
-4
-4.5
Immersed Interface Method
-5
-5
-4
-3
-2
-1
0
Two-Sided Method(IIM)
One-sided method
-5
1
-5
log10(h)
-4
-3
-2
-1
0
log10(h)
Figure 4: (a): Grid refinement analysis of the approximate solution with 14 refinements starting at
n = 20 . Note that the axis has been adjusted so some points may not be visible. The slope of the
regression line fit is 2.1 (b):Grid refinement analysis of the left derivative at the interface with the
one-sided and two-sided(IIM) methods. Note that a linear regression fit shows that the slope of the
line of best fit is about 1.06 and 1.96 for the one-sided and two-sided methods respectively. This
shows the IIM two sided method gave an order 2 accurate approximation while standard one-sided
methods gave a first order accurate approximation.
3
4
Example 3
Consider the problem where

(15)2 (1 + cos(x) ) sinh(15x) − (15) sin(x) cosh(15x) − (1 + x)2 sinh(15x) if
2
2
f (x) =

2
(−(x + 1) cosh(x) − 2x sinh(x)) − ln(2 + x) cosh(x) if x > α.
x≤α
The exact solution can be shown to be
u(x) =

sinh(15x) if
cosh(x)
if
(β ux (x)) x -σ(x)u(x)=f(x) + cδ (x-α) Approximation and Exact Solution with DBC
60
3
x≤α
x>α
×10 -3
Error (difference) Plot
50
2.5
40
2
U-u(x)
u(x)
Exact
Approximation
30
1.5
20
1
10
0.5
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
x
0.5
0.6
0.7
0.8
0.9
1
x
Figure 5: (a): An example of a computed solution with n = 640 grid divisions. Note the actual
solution has its discontinuity connected by the plotter. (b): The error difference of the solution and
actual solution at the grid points.
loglog Plot of U -x (α) Error
Solution Error Plot on a log-log Scale for the IIM
-0.5
-1
0
-1.5
-1
log10( ||U -x(α)-u -x(α)|| ∞ )
log10( ||U-u||∞ )
-2
-2.5
-3
-3.5
-2
-3
-4
-4
-4.5
Immersed Interface Method
-5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
Two-Sided Method(IIM)
One-sided method
-5
-0.5
-5
log10(h)
-4
-3
-2
-1
0
log10(h)
Figure 6: (a): Grid refinement analysis of the approximate solution with 14 refinements starting at
n = 20.Note that the axis has been adjusted so some points may not be visible. The slope of the
regression line fit is 1.96 (b):Grid refinement analysis of the left derivative at the interface with the
one-sided and two-sided(IIM) methods. Note that a linear regression fit shows that the slope of the
line of best fit is about 1.02 and 2.02 for the one-sided and two-sided methods respectively. This
shows the IIM two sided method gave an order 2 accurate approximation while standard one-sided
methods gave a first order accurate approximation.
4
5
Acknowledgments
This project was funded by the following agencies and grants:
NSF DMS-1461148
NSA H98230-15-1-0024
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