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

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.

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 ).

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 ).

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.

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

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.

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

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)

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 :

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.

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.

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