1. Numerical Methods on Partial Differential Equation Md. Mashiur Rahman Department of Physics University of Chittagong Laplace Equation

2. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 02/10 Expansion of a function f(x) about a point x = a in an infinite series of terms involving derivatives of the function f(x) evaluated at that point. Taylor Series : ƒ ( n ) ( a ) denotes the n th derivative of ƒ evaluated at the point x = a. English Mathematician (1685 – 1731) R n ( x ) denotes the Remainder after n terms: [ ]: closed interval ( ): open interval If a = 0, then Taylor series is called Maclaurin series.

3. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 03/10 Expansion of a function f(x+h) about a point x : Taylor Series : O ( h ) denotes the terms involving h and its higher degree. By rearranging, Forward Finite Difference approximation  O ( h ) is known as Local Truncation Error.  Local truncation error is proportional to the step-size ( h ).

4. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 04/10 Expansion of a function f(x – h ) about a point x : Taylor Series : By rearranging, backward Finite Difference approximation  O ( h ) is known as Local Truncation Error.  Local truncation error is proportional to the step-size ( h ).

5. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 05/10 Subtracting f(x + h ) from f(x – h ) : Taylor Series : By rearranging, central Finite Difference approximation  For central approximation, Local Truncation Error is of the order of h 2. For both the forward and backward approximation, it is O(h). So, central approximation gives better approximation for the 1 st order derivative.

6. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 06/10 Geometrical interpretation :  So, there is a huge difference between derivative and finite difference. x f(x) f(x+h) f(x-h) f’(x) Forward:Backward: Central: Derivative: f(x) = slope Smaller the step-size, Better the approximation. x x+h x–h h h

7. Finite Difference Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 07/10 Adding f(x + h ) from f(x – h ) : Double derivative : By rearranging, Double Derivative of FD approximation  Local Truncation Error is of the order of h 2.

8. Finite Difference Method Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 08/10 FDM transforms derivatives into Finite Difference. Derivatives apply for continuous function. Finite Difference requires functional values at different points. So, non-continuous space is enough for FD. This is done dividing the space into number of equal sections. x y h k Discretization Mesh/Grid Mesh-point /Grid-point Mesh-lines /Grid-lines

9. Laplace Equation Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 09/10 This is a 2D problem and so, xy- plane is enough. xy- plane has to be divided into some grids. For simplicity, let grid-sizes in both directions are equal h = k.  Elliptic:  Time independent problem  Solution: u(x, y) = ? x y h h Solutions u(x, y) have to be determined at grid points. x = ih, i = 0, 1, 2, 3, ….. y = jh, j = 0, 1, 2, 3, ….. Grid points: (i, j)

10. Laplace Equation Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 010/10 In the Grid representation: x y h h x = ih, i = 0, 1, 2, 3, ….. y = jh, j = 0, 1, 2, 3, ….. Solutions at grid-point (i, j): Double derivatives :

11. Laplace Equation Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 011/10 x = ih, i = 0, 1, 2, 3, ….. y = jh, j = 0, 1, 2, 3, ….. Laplace equation becomes : In the Grid representation: x y Standard Five-Point Formula (Five-point stencil) Value of u at any grid point is the average of its values at four neighbouring points to left, right, up & down.

12. Laplace Equation Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 012/10 Laplace equation remains invariant under rotation of 45 . Diagonal Five-Point Formula: x y Value of u at any grid point is the average of its values at four diagonal points. Error in Diagonal formula is FOUR TIMES than that in Standard formula. Standard Five-point Formula should be preferred, if possible.

13. Numerical Methods for PDE Md. Mashiur RahmanTuesday, June 09, 2015 Department of Physics, CU. Slide: 013/10 What we have learnt:  Finite Difference Method  Transformation of Laplace Equation in FDM Next class:  Solution of Laplace equation