Question 1:
Consider a steel plate of size 15cm x 15cm. If two of the sides are held at 100 degree celsius and the other two sides at 0 degree celsius, find the steady state temperatures of interior points, assuming a grid size of 5cm x 5cm.

Solution
Using the five point difference formula:
\(
\begin{align*} -4{ f }_{ 1 }+{ f }_{ 2 }+{ f }_{ 3 }+100+100\quad &=\quad 0 \\ -4{ f }_{ 2 }+{ f }_{ 1 }+100+0+{ f }_{ 4 }\quad &=\quad 0 \\ -4{ f }_{ 3 }+{ f }_{ 1 }+100+{ f }_{ 4 }+0\quad &=\quad 0 \\ -4{ f }_{ 4 }+{ f }_{ 2 }+{ f }_{ 3 }+0+0\quad &=\quad 0 \end{align*}
\)
Solving the above system using Gauss Elimination method, we get:

\(
{ f }_{ 1 }=75,{ f }_{ 2 }=50,{ f }_{ 3 }=50{ f }_{ 4 }=25
\)

Note: The linear system in the above problem is diagonally dominant. Therefore, it can be solved by Gauss-seidel method. When such an iteration is applied to Laplace’s equation, the iterative method is called Liebmann’s iterative method.

Question 2:
Solve the above problem using Liebmann’s iterative method.
Solution
Using the standard five point formula,
\(
\begin{align*}{ f }_{ 1 }&=\frac { { f }_{ 1 }+{ f }_{ 3 }+200 }{ 4 } \\ { f }_{ 2 }&=\frac { { f }_{ 1 }+{ f }_{ 4 }+100 }{ 4 } \\ { f }_{ 3 }&=\frac { { f }_{ 1 }+{ f }_{ 4 }+100 }{ 4 } \\ { f }_{ 4 }&=\frac { { f }_{ 2 }+{ f }_{ 3 } }{ 4 } \end{align*}
\)

To Find Initial Approximation
Using diagonal five point formula, assuming \({ f }_{ 4 }=0\), we get:
\(
{ f }_{ 1 }=\frac { 100+{ f }_{ 4 }+100+100 }{ 4 } =75
\)
Using standard five point difference formula:
\(
\begin{align*} { f }_{ 2 }&=\frac { 1 }{ 4 } \left( 75+0+100+0 \right) =\quad 43.75 \\ { f }_{ 3 }&=\frac { 1 }{ 4 } \left( 100+75+0+0 \right) =\quad 43.75 \\ { f }_{ 4 }&=\frac { 1 }{ 4 } \left( 43.75+43.75+0+0 \right) =\quad 21.88 \end{align*}
\)

Now we use Gauss-Siedel iteration method to find better approximation of the solution. Using this, we get the following values:

\({ f }_{ 1 }\) \({ f }_{ 2 }\) \({ f }_{ 3 }\) \({ f }_{ 4 }\)
1st iteration 71.88 48.44 48.44 24.22
2nd iteration 74.22 49.61 49.61 24.81
3rd iteration 74.81 49.90 49.90 24.95
4th iteration 74.95 49.98 49.98 24.99
Solve Laplace Equation Using Finite Difference Method
Solving Laplace Equation using Finite Difference Method – Problem Set – II

Written by Tessy C

Lecturer and Research Scholar in Mathematics.