CE 601: Numerical Methods

Lecture 31

The Classification of PDEs

Course Coordinator:Dr. Suresh A. Kartha,Associate Professor,

Department of Civil Engineering,IIT Guwahati.

The Classification of PDEs

• We discussed about the classification of PDEs for a quasi-linear second order non-homogeneous PDE

as elliptic, parabolic and hyperbolic.

• Such Classification helps in knowing the allowable initial and boundary conditions to a given problem.

• It also helps in the effective choice of numerical methods.

2 2 2

2 2

f f f f fA B C D E Ff G

x x y y x y

• We can also have another general case: First order quasi-linear non-homogeneous PDE in two independent variables:

• We can also have cases involving two general quasi-linear first order non-homogeneous PDE in two independent variables

f fa b c

x y

f f g ga b c d e

t x t x

f f g gA B C D E

t x t x

• These are some of the general PDEs that are extensively used.

• The classification of all the above PDEs are related to the characteristics of PDE.

• What are characteristics of PDE?

• If we consider all the independent variables in a PDE as part of describing the domain of the solution than they are dimensions

• e.g. In

The solution ‘f’ is in the solution domain D(x,t). There are two dimensions x and t.

2

2; ( , )

f ff x t

x x

• E.g.

• Similarly higher dimensions can be present.

• Characteristics are (n-1) dimensional hypersurfaces in n-dimensional domain that have special features.The characteristics are paths in the solution domain along

which information propagates.

Discontinuities in derivatives of the dependent variable if it is there, also propagates through these characteristic paths.

2 2

1 22 2

2 2 2

1 2 32 2 2

( , , ) ( , , ) 3 dimensions

( , , , ) ( , , , ) 4 dimensions

f f f

t x y

f x y t D x y t

f f f f

t x y z

f x y z t D x y z t

• Consider the simple convection problem (say convection of a property ‘f’ of a fluid particle) in one spatial dimension:

• We can see that the solution domain is D(x,t) of two dimensions. Therefore we should try to infer in a domain (x,t) whether there are any characteristics paths.

• Let the convection velocity ‘u’ be given as u(t). A moving fluid particle can carry mass, momentum, energy as it moves through the space. Let convection velocity for that particle be ‘u’.

0f f

ut x

for the particledx

udt

• Or, . This is the pathline of the said fluid particle.

• On integration, it means that ‘f’ will be a constant.

• This means that the property ‘f’ is convected along a pathline.

• This pathline becomes the characteristic path for the property ‘f’ in the solution domain (x,t).

0

0 ( )

t

t

x x u t dt

0f f f dx f df

ut x t dt x dt

• In hydrology, some of you might have studied Kinematic Wave Routing. In that characteristic paths are used to rout incoming flood hydrograph.

To determine Characteristics• Note that discontinuities in the derivatives of the

solution (if exist) must propagate along the characteristics.

• We already have the quasi-linear second order PDE:

• If the solution domain is D(x,y) for the dependent variable f(x,y), then at any general point ‘P’ in the solution domain, if

are multi-valued or

discontinuous and if a path passes through this general point ‘P’, then it is a characteristic path.

2 2 2

2 2 or or

f f f

x x y y

2 2 2

2 2 (1)

f f f f fA B C D E Ff G

x x y y x y

• We can use chain rule

2 2

2

(2)

f f fd dx dy

x x x y

f f fd dx dy

y x y y y

2

2

2

2

2

i.e. 0 (3)

0

f ff D E Ff Gx yx

A B Cf f

dx dy dx y x

dx dyf f

dy y

• In this system if we set the determinant of coefficient matrix = 0

• Eq.(4) becomes the characteristic equation

• If , then we have complex characteristic curves (Elliptic PDE).

• If , then we have equal characteristic curves (Parabolic PDE).

2 4

2

dy B B AC

dx A

2 4 0B AC

2 4 0B AC

2 2

2

i.e. 0

i.e. 0 (4)

A dy B dx dy C dx

dy dyA B C

dx dx

• If , then we have hyperbolic PDE and distinct characteristic paths.

Elliptic PDEs have no real characteristic paths.Parabolic PDEs have one real repeated characteristic path.Hyperbolic PDEs have two real and distinct characteristic paths.

• Due to presence of characteristic paths in the solution domain say D(x,y), we have– Domain of dependence– Range of influence Note in the figures we represent: Horizontal lines as Domain of

dependence; Vertical lines as Range of influence.

2 4 0B AC

• Elliptic PDE:

• Parabolic PDE

• Hyperbolic PDE

• Consider the point P in the solution domain.

• f(xp ,yp) depends on everything that has happened in domain of dependence.

• For a quasi-linear first order non-homogeneous PDE, the PDE is always hyperbolic

• The characteristic paths are determined by

(1)f f

a b ct x

(2)f f

df dt dxt x

• Characteristic equation:

• Eq.(3) is differential equation for a family of paths in the solution domain along which

and may be discontinuous or multi-valued.

• As ‘a’ and ‘b’ are real valued, real characteristic paths always exist.

i.e. a b f t c

dt dx f x df

0

i.e. (3)

adx bdt

dx b

dt a

f

tf

x

• Speed of Propagation

• Speed of propagation of information depends on the slope of the characteristics.

• Physical problems governed by PDEs that have real characteristics are Propagation Problems.

• For elliptic PDE:

• Roots are complex.

• Elliptic PDEs are not having any realcharacteristics.

• The entire solution domain D(x,y) is thedomain of dependence as well as domaininfluence.

2 2

2 20

T T

x y

2 4

2

0 0 4i.e. 1

2

dy B B AC

dx A

dy

dx