Pde Classification

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Computational Mechanics 5 - 1 David Apsley

5. PARTIAL DIFFERENTIAL EQUATIONS SPRING 2005

5.1 Introduction

5.2 Characteristics

5.2.1 First-order equations

5.2.2 Second-order wave equation5.3 Classification of second-order PDEs

5.3.1 Hyperbolic, parabolic and elliptic PDEs

5.3.2 Domain of dependence and boundary conditions

5.1 Introduction For ordinary differential equations (ODEs) two numerical strategies have been considered:

• forward marching – the solution is advanced forward, one step at a time;

• finite differences – derivatives are approximated by algebraic differences and the

solution is obtained for all points simultaneously.

For any given equation, the solution method is determined by the boundary conditions.

It will be shown that essentially the same two types of solution strategy – forward-marching

or all-points-simultaneously – may be employed for partial differential equations (PDEs).

However, for PDEs, the equation itself determines the required boundary conditions and

hence the type of solution method.

It will be shown that certain second-order PDEs (which are very common in science and

engineering) can be categorised as hyperbolic, parabolic or elliptic, and that these have very

different types of solution and boundary conditions.

Hyperbolic and parabolic equations are solved by forward-marching techniques – advancing

forwards in the direction of one independent variable; (in real applications this is usually the

time, t ). Thus, these are often referred to as time-marching methods. Elliptic equations arise

where there is no natural preferred direction of propagation; the independent variables are the

physical space coordinates x and y. In these cases, finite-difference methods are used to solve

simultaneously for all points in the domain.

5.2 Characteristics

PDEs are more complex than ODEs because they contain derivatives with respect to more

than one variable. Wouldn’t it be nice if some equation manipulation would allow them to be

reduced to ordinary differential equations? Some equations are, indeed, amenable to such

analysis.

Definition: A characteristic is a curve along which a partial differential equation reduces to

an ordinary differential equation.




5.2.1 First-Order Equations

Example. First-order wave equation with constant wave speed c.

0on)(,0 0 ===∂

∂+

∂

∂t xuu

x

uc

t

u

Suppose that one follows some particular path x(t ). Then, along that line, u = u( x(t ),t ) is a

function of t only and its total derivative with respect to t is

t

x

x

u

t

u

t

u

d

d

d

d

∂

∂+

∂

∂=

Compare with the original differential equation. This is exactly the same as the LHS provided

that the chosen path satisfies

ct

x=

d

d

orconst ct x += .

Along these lines,

0d

d=

t

u (an ordinary differential equation)

Hence, u propagates unchanged along the lines

constant ct x =−≡ . Physically, a wave pulse propagates unchanged with wave speed c.

(Note that, although we have written a total derivative du /dt for that particular path, u is

actually a function of both t and the characteristic . Here, = x0, the x value at t = 0.)

Example. First-order wave equation with amplitude-dependent wave speed.

0on)(,0 0 ===∂

∂+

∂

∂t xuu

x

uu

t

u

Along any line x(t ), the total derivative of u is

t

x

x

u

t

u

t

u

d

d

d

d

∂

∂+

∂

∂=

Comparing this with the original equation, choose

lines such that d x /dt = u. Then

ut

x

t

u==

d

dalong0

d

d.

i.e. u propagates unchanged along lines ut x x += 0 .

Note that larger-amplitude waves travel faster; in general such waves will steepen (and

eventually break).

t

xx0

x +ct

ξ = c o n s t a n t

0

t

xx0

x +ut

characteristics

0




Example

22 )0,(,3 x xu y x y

u x

x

u=+=

∂

∂+

∂

∂

Find u(3,19).

Solution

We require a path y( x) which will take us from the x-axis

where the boundary conditions are known to the point

(3,19).

Along any curve y( x), the total derivative of u is

x

y

y

u

x

u

x

u

d

d

d

d

∂

∂+

∂

∂= .

By comparing with the given equation, one should choose

the path such that 23 /dd x x y = . Hence, the characteristicsare the set of curves

C x y += 3

The particular characteristic passing through the point (3,19)

is

83 −= x y

Along this characteristic,

8d

d

3 −+=

+=

x x

y x x

u

and hence, along this characteristic, u varies as

C x x xu +−+= 84

412

21

The curve 83 −= x y cuts the x axis at x = 2; at this point the boundary conditions give u = 4.

Substituting these values fixes C = 14. Hence, along the characteristic passing through (3,19)

u is given by

1484

412

21 +−+= x x xu

Thus, the value when x = 3 is

459]148[)19,3( 3

4412

21 =+−+= = x x x xu

Exercise for the willing: show that the general solution is3 / 43

433 / 23

2134

412

21 )()()( y x y x x y x x xu −+−+−++=

(2,0)

(3,19)

characteristics

y

x

u known




General Method of Characteristics For 1st-Order PDEs

The general quasi-linear 1st-order PDE is

c y

ub

x

ua =

∂

∂+

∂

∂

where a, b, c are functions of x, y, u, and the dependent variable u is given on some initial

curve.

Write the equation as

a

c

a

b

y

u

x

u=

∂

∂+

∂

∂

Now, for any curve y( x):

x

y

y

u

x

u

x

u

d

d

d

d

∂

∂+

∂

∂=

Hence if one chooses characteristic curves y( x) such that a

b

x

y

=d

d, then u will evolve along

these curves according toa

c

x

u=

d

d.

All first-order partial differential equations of the form

c y

ub

x

ua =

∂

∂+

∂

∂

(where a, b and c may be functions of x, y and u).have a formal solution and characteristics

given by

ab

x y =dd ,

ac

xu =

dd

For completely, general a, b and c these are simultaneous ordinary differential equations,

which may be solved by the forward-marching techniques of section 2.

The solution and characterised are sometimes summarised as

c

u

b

y

a

x ddd==




5.2.2 Second-Order Wave Equation

Under certain circumstances (“hyperbolic” equations – see later) 2nd

-order PDEs in the plane

may be “factorised” in such a way as to be soluble by the method of characteristics.

Second-order wave equation:

01

2

2

22

2

=∂

∂−

∂

∂

t

u

c x

u

If c (we shall see later that this is a wave-speed ) is constant, this can be factorised as

0)1

)(1

( =∂

∂−

∂

∂

∂

∂+

∂

∂u

t c xt c x (*)

We attempt to choose and such that

∂

∂+

∂

∂=

∂

∂

∂

∂−

∂

∂=

∂

∂

t c xt c x

1

2

1,

1

2

1

(The factors of ½ aren’t strictly necessary – they happen to be convenient for what follows.)

This can readily be achieved by taking

)(2

1,)(

2

1−=+=

ct x

which can be inverted to give

ct xct x +=−= ,

Substituting into (*) reduces the wave equation to the simple form

0

2

=∂∂

∂ u

This may now be integrated twice (first with respect to , along = constant ; then with

respect to along = constant ):

)(d)(

)(

G f u

f u

+

=

=∂

∂

(where f ( ) and G( ) are arbitrary). Writing

= d)()( f F , the

general solution of the 2-d second-order wave equation is)()( ct xGct xF u ++−=

The solution consists of a right-travelling wave ( constant ct x =− )

and a left-travelling wave ( constant ct x =+ ). F and G can be any

functions. Their precise form will be fixed by appropriate boundary

conditions – typically initial boundary conditions: u and ∂u / ∂t

specified everywhere at t = 0.

The key point is that, for this type of equation, information propagates along characteristics.

This must be reflected in the discretisation scheme.

x

t

(x,t)

x - c t =

c o n s t

x + c t = c o n s t

zone of influence

initial data

zone of dependence




5.3 Classification of Second-Order PDEs

5.3.1 Hyperbolic, Parabolic and Elliptic PDEs

An equation of the form

f y

c y x

b x

a =∂φ∂+

∂∂φ∂+

∂φ∂

2

22

2

2

where a, b, c and f may be functions of x, y, φ, ∂φ / ∂ x and ∂φ / ∂ y, but not the second

derivatives, is called quasi-linear .

The nature of the differential equation and its analytical or numerical solution is determined

by the value of the discriminant acb 42 − .

For constant a, b and c this relates to the ability to factorise it into two first-order

differentials, each of which will have its set of characteristics along which the solution can be

forward-marched. The general analysis is a bit more complex.

The cases are:

042 >− acb hyperbolic two sets of characteristics forward-marching solution

042 =− acb parabolic one set of characteristics forward-marching solution

042 <− acb elliptic no characteristics simultaneous solution for all points

If a, b or c is not constant then it is possible for the governing equations to be of different

types in different parts of the domain.

Examples

(i) φ=∂∂

φ∂+

∂

φ∂+

∂

φ∂343

2

2

2

2

2

y x y x

a = 1, b = 4, c = 3;

(b2 – 4ac = 4)

hyperbolic

(ii) 02

2

=∂

φ∂+

∂

φ∂+

∂

φ∂+

∂

φ∂

∂

∂

y y y x x

a = 1, b = 1, c = 1;

(b2 – 4ac = –3)

elliptic

(iii) φ=∂

φ∂+

∂

φ∂2

2

2

2

x y

y x

a = y, b = 0, c = x;

(b2 – 4ac = –4 xy)

elliptic in 1st /3

rd quadrants,

hyperbolic in 2nd /4

th quadrants




5.3.2 Domain of Dependence and Boundary Conditions

The nature of the equation determines the required

boundary conditions. For elliptic equations, values of

φ or its derivative are required on all boundaries of the

domain. Forward-marching schemes are like initial-value problems in ODEs: For a second-order equation

this means that we require φ and its forward derivative

on the initial boundary.

Type Example Equation Application(s)

Hyperbolic Wave equation

01

2

2

22

2

=

∂

φ∂−

∂

φ∂

t c x

Vibrating string: c2 = T /

φ = displacement;

T = tension;

= mass per unit length.

Longitudinal vibration in a rod: c2 = E /

φ = displacement;

E = Young’s modulus;

= density.

Parabolic Diffusion equation2

2

xt ∂

φ∂=

∂

φ∂

Heat equation: = k/ cP

φ = temperature;

k = thermal conductivity;

= density;

cP = specific heat capacity.

Consolidation equation:φ = excess pore pressure.

= coefficient of consolidation.

Elliptic

Poisson equation

(Laplace equation

if RHS = 0)

f y x

=∂

φ∂+

∂

φ∂2

2

2

2

Steady-state distributions such as:

temperature distribution;

electrostatic potential;

stress distribution;

velocity potential.

(Note the subtle change between t and y for the second independent variable: the former tends

to occur in time-dependent problems).

t

t

y

x

x

x

hyperbolic

parabolic

elliptic

Domain of dependence for

different types of PDE

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Pde Classification

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