EXAMPLE: Solve the Laplace's equation on the following small grid with Ax = Ay using Jacobi Iteration with tol=5 x 10-3. Set starting values to zero for all interior points. 1₂.3 1₁,2 21.4 8.9 8.4 7.2 6.1 14₁.1 142,4 8.9 ₁3 6.8 142.1 8.9 1₁.2 7.7 23.1 1844 8.9 W 1143 8.7 14.1 15,4 8.9 9.2 9.4 9.8 15.1 15,3 U5.2
EXAMPLE: Solve the Laplace's equation on the following small grid with Ax = Ay using Jacobi Iteration with tol=5 x 10-3. Set starting values to zero for all interior points. 1₂.3 1₁,2 21.4 8.9 8.4 7.2 6.1 14₁.1 142,4 8.9 ₁3 6.8 142.1 8.9 1₁.2 7.7 23.1 1844 8.9 W 1143 8.7 14.1 15,4 8.9 9.2 9.4 9.8 15.1 15,3 U5.2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:7-Laplace's equation-Iterative method.... >>>
Q
Jacobi Iteration:
m
m
m
uj-1 + u-1₁j + ui+1) + uj+1
4
Note that this formula assumes Ax = Ay.
For each interior grid point (i, j), uij at the next iteration (m+1) is found
from (6). Once an iteration has been completed for all interior grid points,
we compute the infinity norm for vector um+1- um
U₁₁jm+1 =
If lum+1 um < tol, where tol is a pre-defined tolerance, the
iterations terminate and the solution to (5) is um+1, otherwise the
iterations continue.
To start the iteration (m-0) values must be given for the unknown interior
values of uij. These values can be set to zero or interpolated from the
known boundary values.
EXAMPLE: Solve the Laplace's equation on the following small grid
with Ax = Ay using Jacobi Iteration with tol-5 x 10-³. Set starting
values to zero for all interior points.
11,3
m+1 =
ui.j
11,2
Jacobi Iteration:
U14
u¹= (U2,1"
22,2
m+1
8.9
8.4
m
7.2
6.1
241.1
22,4
8.9
m
13
U₁
6.8
142.1
U3,4
8.9
m
m
m
uj-1 + U-1₁j + U+1.j + ulj +1
4
for all interior points we have:
th
ULT
7.7
13.1
U44
8.9
1143
143
8.7
772
m
= (U2,1 + U₁,2" + U3,2+U2,3)/4
14.1
...
USA
8.9
9.2
9.4
۱۲:۱۱
9.8
15.1
(6)
(6)
US,3
Us,2
A
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
