COMPUTATIONAL MODELING FOR ENGINEERINGMECN 6040

Professor: Dr. Omar E. Meza Castilloomeza@bayamon.inter.edu

http://facultad.bayamon.inter.edu/omezaDepartment of Mechanical Engineering

FINITE DIFFERENCE METHODS FOR HYPERBOLIC PROBLEMS

Wave Equation

PARTIAL DIFFERENTIAL

EQUATIONS▪ Our study of PDE’s is focused on linear,

second-order equations

▪ The following general form will be evaluated for B2 - 4AC (Variables – x and y/t)

A2ux 2

B2uxy

C2uy 2

D0

B2-4AC Category Example

< 0 Elliptic Laplace equation (steady state with 2 spatial dimensions)

= 0 Parabolic Heat conduction equation (time variablewith one spatial dimension)

>0 Hyperbolic Wave equation (time-variable with onespatial dimension)

2u t 2

c 22ux 2

2ux 2

2uy 2

0

ut

2ux 2

PARTIAL DIFFERENTIAL

EQUATIONS

WAVE EQUATION

Mechanical vibrations of a guitar string, or in the membrane of a drum, or a cantilever beam are governed by a partial differential equation, called the Wave Equation.

To derive the wave equation we consider an elastic string vibrating in a plane, as the string on a guitar. Assume u(x,t) is the displacement of the string away from its equilibrium position u=0.

u(x,t)

x

u

T(x+Dx,t)

T(x,t)

x x+Dx

a

b

WAVE EQUATION

Newton’s Law: F=ma

Vertical:

where ρ is the mass density function of the string

Horizontal:

As we assume there is only vertical motion of the string

T(x+Dx,t)

T(x,t)

a

b

FVert T(x x)sin T(x,t)sin

ma x2ut 2

FHoriz T(x x)cos T(x, t)cos 0

WAVE EQUATION

Let

Then,

So, Fvert = ma implies, from the previous slide, that

Or,

T T(x x)cos T(x,t)cos

T(x x) T

cos, T(x, t)

T

cos

T

cossin

T

cossin x

2ut 2

T(tan tan) x2ut 2

WAVE EQUATION

Now, tan β is just the slope of the

string at x+Δx and tan a is the

slope at x.

Thus,

So,

T(x+Dx,t)

T(x,t)

a

b

tan ux xx

and tan ux x

T(ux xx

ux x

) x2ut 2

1

x(ux xx

ux x

) T

2ut 2

WAVE EQUATION

Now, take the limit as Δx -> 0, to get

This the Wave Equation in one dimension (x)

1

x(ux xx

ux x

) T

2ut 2

2ut 2

c 22ux 2

WAVE EQUATION

WAVE EQUATIONBOUNDARY CONDITIONS

Since the wave equation is second order in space and time, we need two boundary conditions for each variable. We will assume the string x parameter varies from 0 to L.

Typical set-up is to give boundary conditions at the ends of the string in time:

u(0,t) = 0 u(1,t) = 0

and space initial conditions for the displacement and velocity:

u(x,0) = f(x) ut(x,0)=g(x)

00=

)()0,(

)()0,(

0),(

0),0(

02

xgxu

xfxu

tLu

tu

tLxucu

t

xxtt

WAVE EQUATIONBOUNDARY CONDITIONS

ONE DIMENSIONALWAVE EQUATION

▪ We will solve this equation for x and t values on a grid in x-t space:

t

x00 x ix

nxh /1

jt

00 t

mTtk /

mtT

1nx

Approximatesolution

uij=u(xi, tj) at grid points

FINITE DIFFERENCE APPROXIMATION

▪ To approximate the solution we use the finite difference approximation for the two derivatives in the wave equation. We use the centered-difference formula for the space and time derivatives:

),(),(2),(),(

),(),(2),(1

),(

112

22

112

jijijijixx

jijijijitt

txutxutxuh

ctxuc

txutxutxuk

txu

FINITE DIFFERENCE APPROXIMATION

▪ Then the wave equation (utt =c2 uxx ) can be approximated as

Or,

Let p = (ck/h) Solving for ui,j+1 we get:

1

k 2ui, j 1 2ui, j ui, j1 c

2

h2ui 1, j 2ui, j ui1, j

ui, j 1 p2ui1, j 2(1 p2)uij p

2ui1, j ui, j 1

ui, j 1 2ui, j ui, j1 k2c 2

h2ui 1, j 2ui, j ui1, j

ONE DIMENSIONAL WAVE EQUATION

▪ Note that this formula uses info from the j-th and (j-1)st time steps to approximate the (j+1)-st time step:

t

x

1ix ix 1ix

1jt

jt

mtT

00 t

00 x 1nx

ONE DIMENSIONAL WAVE EQUATION

▪ The solution is known for t=0 (j=0) but the formula uses values for j-1 , which are unknown. So, we use the initial condition ut(x,0)=g(x) and the centered difference approximation for ut to get

▪ At the first time step, we then get

)(22

)()0,( 1,1,1,1,

iiiii

iit xkguuk

uuxgxu

))(2()1(2 1,0,12

02

0,12

1, iiiiii xkguupupupu

ONE DIMENSIONAL WAVE EQUATION

▪ Simplifying, we get

▪ So,

)(5.0)1(5.0 0,12

02

0,12

1, iiiii xkgupupupu

1,,1

22,1

21,

0,12

02

0,12

1,

)1(2

)(5.0)1(5.0

jijiijjiji

iiiii

uupupupu

xkgupupupu