KTU: ME305 : COMPUTER PROGRAMMING & NUMERICAL METHODS : 2017
Module: VI : Solution of Partial Differential Equations: Laplace equation, Finite Difference Method.

Laplace equation is \(\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }f }{ \partial { y }^{ 2 } } =0\)
Replacing second order derivatives by their finite difference equivalents at the point \(({ x }_{ i }{ y }_{ j })\),

\(
{ \left( \frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }f }{ \partial { y }^{ 2 } } \right) }_{ i,j }=\frac { { f }_{ i+1,j }-2{ f }_{ i,j }+{ f }_{ i-1,j } }{ { h }^{ 2 } } +\frac { { f }_{ i,j+1 }-2{ f }_{ i,j }+{ f }_{ i,j-1 } }{ { k }^{ 2 } } =0
\)

For simplicity, consider h=k. Then,

\(
{ f }_{ i+1,j }-2{ f }_{ i,j }+{ f }_{ i-1,j }+{ f }_{ i,j+1 }-2{ f }_{ i,j }+{ f }_{ i,j-1 }=0\\ i.e,{ f }_{ i,j }=\frac { 1 }{ 4 } \left[ { f }_{ i+1,j }+{ f }_{ i-1,j }+{ f }_{ i,j+1 }+{ f }_{ i,j-1 } \right]
\)



This equation contains four neighboring points around the central point \(({ x }_{ i }{ y }_{ j })\) and is known as the five point difference formula for Laplace’s equation.

The Laplace’s equation remains invariant even after rotation of the co-ordinate axes through \({ 45 }^{ o }\). Hence the above equation can be written as:

\(
{ f }_{ i,j }=\frac { 1 }{ 4 } \left[ { f }_{ i-1,j+1 }+{ f }_{ i-1,j-1 }+{ f }_{ i+1,j-1 }+{ f }_{ i+1,j+1 } \right]
\)

This is called standard diagonal five point formula. These formulas will lead to approximate values at the internal grid points, when the boundary conditions are given.
To obtain better approximation, iterative methods are to be adopted.
Following are some of the iterative methods used with finite method approximation.

  • Jacobi method
  • Gauss-seidel or Liebman’s method
  • Relaxation method
Finite Difference Method : Solve Partial Differential Equations
Solving Laplace Equation using Finite Difference Method - Solved Problem - I

Written by Tessy C

Lecturer and Research Scholar in Mathematics.