KTH Course: Applied Numerical methods Team members

Professor: Lennart Edsberg Andreas Angelou

Paul Evans

Vasileios Papadimitriou

Daniel Tepic

Dionysios Zelios

I.C : ( ,0) 0u x ( ,0) 0u

xt

0 x l

0 x l

B.C : X=0

X=L

2 2

2 2

1u R u u

t L dt LC x

1( , ) ( , ) 0

u ul t l t

t xLC

0 ( ) 1u t [V]

0.004[ ]R 610 [ ]L H 80.25 10 [ ]C F

(N+1) grid-points

Discretization of x-axis into N intervals:

Finite Difference Method

first and second order derivatives:

Central difference approximations

Approximation of the right end of B.C:

Upwind discretization (FTBS)

1 1 1 1

1 1

2 2

2 21

2

n n n n n n n n

i i i i i i i iR

t L t LC x

u u u u u u u u

1

1 1

1( 2 ( 1) )

1

n n n n

i i i i

a

au u u u

2

2

1 ( )

( )

ta

LC x

2

t R

L

65 10t s

N=100 grid-points

N=100 grid-points

N=100 grid-points

N=100 grid-points

62.5 10t s

N=200 grid-points

N=200 grid-points

N=200 grid-points

N=200 grid-points

Taylor expansion

N=100-

N=200

Greater interval for finding stability

Without Taylor expansion

N=100-

N=200

Experimental values

N=100-

N=200

Approximation method stability max

N=100

N=200

65.025 10t 62.506 10t

67.0711 10t 63.5355 10t

64.9962 10t 62.4991 10t

65 10t 62.5 10t

N=100 grid-points

N=100 grid-points