1. PARTIAL DIFFERENTIAL EQUATIONS
ENGR 351
Numerical Methods for Engineers
Southern Illinois University Carbondale
College of Engineering
Dr. L.R. Chevalier
Dr. B.A. DeVantier

2. Copyright© 2000 by Lizette R. Chevalier
Permission is granted to students at Southern Illinois University at Carbondale
to make one copy of this material for use in the class ENGR 351, Numerical
Methods for Engineers. No other permission is granted.
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3. Partial Differential Equations
An equation involving partial derivatives of an unknown function of two or more independent variables
The following are examples. Note: u depends on both x and y

4. Partial Differential Equations
An equation involving partial derivatives of an unknown function of two or more independent variables
The following are examples. Note: u depends on both x and y

5. Partial Differential Equations
Because of their widespread application in engineering, our study of PDE will focus on linear, second-order equations
The following general form will be evaluated for B2 - 4AC

6. Partial Differential Equations
Because of their widespread application in engineering, our study of PDE will focus on linear, second-order equations
The following general form will be evaluated for B2 - 4AC

7. B2-4AC Category Example
< Elliptic Laplace equation (steady state with
2 spatial dimensions
= Parabolic Heat conduction equation (time variable
with one spatial dimension
>0 Hyperbolic Wave equation (time-variable with one
spatial dimension

8. y or t
set up a grid
estimate the dependent
variable at the center x

9. y or t
set up a grid
estimate the dependent
variable at the center
or intersections
of the grid x

10. Scope of Lectures on PDE
Finite Difference: Elliptic
The Laplace Equation
Finite difference solution
Boundary conditions
Finite Difference: Parabolic
Heat conduction
Explicit method
Simple implicit method

11. Specific Study Objectives
Recognize the difference between elliptic, parabolic, and hyperbolic PDE
Recognize that the Liebmann method is equivalent to the Gauss-Seidel approach for solving simultaneous linear algebraic equations
Recognize the distinction between Dirichlet and derivative boundary conditions

12. Specific Study Objectives
Know the difference between convergence and stability of parabolic PDE
Understand the difference between explicit and implicit schemes for solving parabolic PDE

13. Finite Difference: Elliptic Equations B2- 4AC < 0
Typically used to characterize steady-state boundary value problems
Before solving, the Laplace equation will be solved from a physical problem

14. Elliptic Equations B2- 4AC < 0
Typically used to characterize steady-state boundary value problems
Before solving, the Laplace equation will be solved from a physical problem

15. The Laplace Equation
Models a variety of problems involving the potential of an unknown variable
We will consider cases involving thermodynamics, fluid flow, and flow through porous media

16. The Laplace equation
Let’s consider the case of a plate heated from the boundaries
How is this equation derived from basic concepts of continuity?
How does it relate to flow fields?

17. Consider the plate below, with thickness Dz.
The temperatures are known at the boundaries.
What is the temperature throughout the plate?
T = 400
T = 200
T= 200
T = 200

18. First, recognize how the shape can be set in an x-y coordinate system

19. Divide into a grid, with increments by Dx and Dy

20. What is the temperature here, if using a block centered scheme?y
T = 400
T = 200
T= 200
T = 200 x

21. What is the temperature here, if using a grid centered scheme?y
T = 400
T = 200
T= 200
T = 200 x

22. Consider the element shown below on the face of
a plate D z in thickness.
The plate is illustrated everywhere by at its edges or
boundaries, where the temperature can be set. y
D y x
D x

23. q(y + D y)
q(x)
q(x + D x)
Consider the steady state heat flux q
in and out of the elemental volume.
q(y)
By continuity, the flow of heat in must equal the flow of heat
out.

24. Rearranging terms

25. Again, this is our continuity equation
Dividing by Dx, Dy, Dz and D t :
As x & y approach zero, the equation reduces to:
Again, this is our continuity equation

26. Equation A
The link between flux and temperature is provided by Fourier’s Law of heat conduction
Equation B
where qi is the heat flux in the direction i.
Substitute B into A to get the Laplace equation

27. Equation A
Equation B

28. Consider Fluid Flow
In fluid flow, where the fluid is a liquid or a gas, the
continuity equation is:
The link here can by either of the following sets of equations:
The potential function:
Stream function:

29. The Laplace equation is then

30. Flow in Porous Media Darcy’s Law
The link between flux and the pressure head is
provided by Darcy’s Law

31. For a case with sources and sinks within the 2-D
domain, as represented by f(x,y), we have the
Poisson equation.
Now let’s consider solution techniques.

32. Evaluate these equations based on the grid and
central difference equations
(i,j+1)
(i,j)
(i-1,j)
(i+1,j)
(i,j-1)

33. we can collect the terms to get:
(i,j)
(i+1,j)
(i-1,j)
(i,j+1)
(i,j-1)
If D x = D y
we can collect the terms
to get:

34. This equation is referred to as the Laplacian difference equation.
(i,j)
(i+1,j)
(i-1,j)
(i,j+1)
(i,j-1)
This equation is referred
to as the Laplacian difference
equation.
It can be applied to all
interior points.
We must now consider
what to do with the
boundary nodes.

35. Boundary Conditions
Dirichlet boundary conditions: u is specified at the boundary
Temperature
Head
Neumann boundary condition: the derivative is specified qi
Combination of both u and its derivative (mixed boundary condition)

36. The simplest case is where the boundaries are
specified as fixed values.
This case is known as the Dirichlet boundary
conditions. u2 u1 u3 u4

37. Consider how we can deal with the lower node shown, u1,1
Note:
This grid would result
in nine simultaneous
equations. u1
1,2 u3
1, ,1 u4
-4u1,1 +u1,2+u2,1+u1 +u4 = 0

38. Let’s consider how to model the Neumann boundary condition
centered finite divided difference
approximation
(0,100)
(0,0)
(200,0)
(200,100)
h = 0.05x + 100 y x
suppose we wanted to consider this end
grid point

39. The two boundaries are consider to be symmetry lines due to the fact that the BC translates in the finite difference form to:
h i+1,j = h i-1,j
and
h i,j+1 = h i,j-1
1,2
1, ,1

40. h1,1 = (2h1,2 + 2 h2,1)/4
h1,2 = (h1,1 + h1,3+2h22)/4
2,2
1,2
1, ,1

41. EXAMPLE
The grid on the next slide is designed to solve the LaPlace equation
Write the finite difference equations for the nodes (1,1), (1,2), and (2,1). Note that the lower boundary is a Dirichlet boundary condition, the left boundary is a Neumann boundary condition, and Dx = Dy.

42. The Liebmann Method
Most numerical solutions of the Laplace equation involves systems that are much larger that the general system we just evaluated
Note that there are a maximum of five unknown terms per line
This results in a significant number of terms with zero’s

43. The Liebmann Method
In addition to the fact that they are prone to round-off errors, using elimination methods on such sparse system waste a great amount of computer memory storing zeros
Therefore, we commonly employ approaches such as Gauss-Seidel, which when applied to PDEs is also referred to as Liebmann’s method.

44. The Liebmann Method
In addition the equations will lead to a matrix that is diagonally dominant.
Therefore the procedure will converge to a stable solution.
Over relaxation is often employed to accelerate the rate of convergence

45. As with the conventional Gauss Seidel method, the iterations
are repeated until each point falls below a pre-specified
tolerance:

46. Groundwater Flow Example

47. Modeling 1/2 of the system shown, we can develop the following
schematic where D x = D y = 20 m
(0,100)
(0,0)
(200,0)
(200,100)
h = 0.05x + 100 y x
The finite difference equations can be solved using a
a spreadsheet. This next example is part of the PDE example you can download from my homepage.

48. Secondary Variables
Because its distribution is described by the Laplace equation, temperature is considered to be the primary variable in the heated plate problem
A secondary variable may also be of interest
In this case, the second variable is the rate of heat flux across the place surface

49. FINITE DIFFERENCE
APPROXIMATION
BASED ON RESULTS
OF TEMPERATURE DISTRIBUTION

50. THE RESULTING FLUX IS A VECTOR WITH MAGNITUDE AND DIRECTION
Note:
q is in degrees
If qx=0, q is 90 or 270  depending on whether qy is positive or negative, respectively

51. Finite Difference: Parabolic Equations B2- 4AC = 0
These equations are used to characterize
transient problems.
We will first study this in one spatial direction
then we will discuss the results in 2-D.

52. Consider the heat-conduction equation
As with the elliptic PDEs, parabolic equations can be solved
by substituting finite difference equations for the
partial derivatives.
However we must now consider changes in time
as well as space

53. y x t x {
temporal {
spatial

54. Forward finite divided
difference
Centered finite divided difference

55. We can further reduce the equation:
NOTE:
Now the temperature
at a node is estimated
as a function of
the temperature at the
node, and surrounding
nodes, but at a previous time

56. EXAMPLE Consider a thin insulated rod 10 cm long with k = 0.835 cm2/s
Let D x = 2 cm and D t = 0.1 sec.
At t=0 the temperature of the rod is zero.
hot
cold

57. Convergence and Stability
Convergence means that as D x and D t approach zero, the results of the numerical technique approach the true solution
Stability means that the errors at any stage of the computation are attenuated, not amplified, as the computation progresses
The explicit method is stable and convergent if

58. Derivative Boundary ConditionsTo TL
In our previous example To and TL were constant values.
However, we may also have derivative boundary conditions
Thus we introduce an imaginary point at i = -1
This point provides the vehicle for providing the derivative BC

59. Derivative Boundary Conditions
q0 = 0 TL
For the case of qo = 0, so T-1 = T1 .
In this case the balance at node 0 is:

60. Derivative Boundary Conditions
q0 = 10 TL
For the case of qo = 10, we need to know k’ [= k/(C)].
Assuming k’ =1, then 10 = - (1) dT/dx,
or dT/dx = -10

61. Derivative Boundary Conditions

62. Implicit Method Explicit methods have problems relating to stability
Implicit methods overcome this but at the expense of introducing a more complicated algorithm
In this algorithm, we develop simultaneous equations

63. Explicit

64. Explicit Implicit

65. Explicit Implicit
grid point involved with space difference

66. Explicit Implicit
grid point involved with space difference
grid point involved with time difference

67. Explicit Implicit
grid point involved with space difference
grid point involved with time difference
With the implicit method, we develop a set of simultaneous
equations at step in time

68. which can be expressed as:
For the case where the temperature level is given at the end
by a function f0 i.e. x = 0

69. Substituting
In the previous example problem, we get a 4 x 4 matrix
to solve for the four interior nodes for each time step