Faculty Name Prof. A. A. Saati Umm Al-Qura University Slide 2 2
Slide 3 Finite Difference Method Parabolic Elliptic Hyperbolic
Partial Differential Equations CH3 3 Umm Al-Qura University Slide 4
Linear Second order PDEs are important sets of equations that are
used to model many systems in many different fields of science and
engineering. Classification is important because: Each category
relates to specific engineering problems. Different approaches are
used to solve these categories. Classification in lecture Part 2,
Part 3 & Part 4.1show that 2 nd Order Linear PDEs as: Elliptic
if Parabolic if Hyperbolic if 4 Finite Difference Method Slide 5
Parabolic Partial Differential Equations Introductory Remarks
Equations of science and engineering such as motion in fluid
mechanics are frequently reduced to parabolic formulation Example:
Boundary layer equation Parabolized Navier-Stokes (PNS) equation
Unsteady heat conduction equation Various finite difference of the
model Parabolic Differential Equation (PDE) will be investigated. 5
Finite Difference Method Slide 6 Finite Difference Formulations
Consider 1-D 2 nd order PDE ( unsteady heat conduction equation )
The model equation has the following form Various finite difference
approximations can be used to represent the derivatives in the
equation Let forward finite difference for of order and central
finite difference for of order 6 Finite Difference Method Slide 7
Finite Difference Formulations [ Explicit Method ] The equation can
be approximated by the following difference equation. In this
equation, is the only unknown and therefore, it can be computed
from the following and the time level n is known form a previous
solution or given as initial data. The 2 nd -order PDE has been
replaced by algebraic equation. 7 Slide 8 8 Slide 9 9 Slide 10
Finite Difference Formulations [Im plicit Method ] When a
first-order backward difference approximation for the time
derivative And a second-order central difference approximation for
the spatial derivative is used The equation takes the form: There
are three unknowns: 10 Slide 11 Finite Difference Formulations [Im
plicit Method ] The computation of the unknowns would require a set
of coupled finite difference equations The above equation can be
rearranged as: A formulation of this type, which includes more than
one unknown in each FDE, is known as an implicit method. The
equation may expressed in the general form as: This FDE is written
for all grid points, resulting in a set of algebraic equations,
These equation are put in a matrix form. 11 Slide 12 Explicit
Methods This section introduces some of the commonly used explicit
methods for solving parabolic equations. 12 Finite Difference
Method Slide 13 The Forward Time/Central space (FTCS) method This
is read truncation error of order The method is stable for 13 Slide
14 The Richardson method In this approximation, central
differencing is used for both time and space derivatives. - this is
read truncation error of order - The method is unconditionally
unstable 14 Slide 15 The DuFort-Frankel method This method is a
modification of the Richardson method - this is read truncation
error of order The equation can be solve explicitly for at time
level The method is unconditionally stable 15 Slide 16 The grid
point involved in the method are shown in the figure. 16 Slide 17
17 Example - 1 (Explicit Methods) x ice Slide 18 18 Example - 1
(Explicit Methods) Slide 19 19 Example - 1 (Explicit Methods) t=0
t=0.25 t=0.5 t=0.75 t=1.0 x=0.25x=0.5 x=0.0 x=0.75 x=1.0 0 0 0 0 0
0 0 0 0 0 Sin(0.25)Sin(0. 5)Sin(0.75) Slide 20 20 Example - 1
(Explicit Methods) t=0 t=0.25 t=0.5 t=0.75 t=1.0 x=0.25x=0.5 x=0.0
x=0.75 x=1.0 0 0 0 0 0 0 0 0 0 0 Sin(0.25)Sin(0. 5)Sin(0.75) Slide
21 21 Example - 1 (Explicit Methods) t=0 t=0.25 t=0.5 t=0.75 t=1.0
x=0.25x=0.5 x=0.0 x=0.75 x=1.0 0 0 0 0 0 0 0 0 0 0 Sin(0.25)Sin(0.
5)Sin(0.75) Slide 22 22 Example - 1 (Explicit Methods) Remarks on
Example 1 Slide 23 23 Example - 1 (Explicit Methods) t=0 t=0.025
t=0.05 t=0.075 t=0.10 x=0.25x=0.5 x=0.0 x=0.75 x=1.0 0 0 0 0 0 0 0
0 0 0 Sin(0.25)Sin(0. 5)Sin(0.75) Slide 24 24 t=0 t=0.025 t=0.05
t=0.075 t=0.10 x=0.25x=0.5 x=0.0 x=0.75 x=1.0 0 0 0 0 0 0 0 0 0 0
Sin(0.25)Sin(0. 5)Sin(0.75) Example - 1 (Explicit Methods) Slide 25
25 Example - 1 (Explicit Methods) t=0 t=0.025 t=0.05 t=0.075 t=0.10
x=0.25x=0.5 x=0.0 x=0.75 x=1.0 0 0 0 0 0 0 0 0 0 0 Sin(0.25)Sin(0.
5)Sin(0.75) Slide 26 26 Implicit Methods Slide 27 When a
first-order backward difference approximation for the time
derivative And a second-order central difference approximation for
the spatial derivative is used The equation takes the form: A
formulation of this type, which includes more than one unknown in
each FDE, is known as an implicit method. 27 Slide 28 Implicit
methods offer great advantage on the stability, since most are
unconditionally stable * A larger step size in time is permitted *
An increase in time step will increase the truncation error. 28
Slide 29 The Laasonen Method When a first-order backward difference
approximation for the time derivative And a second-order central
difference approximation for the spatial derivative is used The
equation takes the form: - this is read truncation error of order
29 Slide 30 The Crank-Nicolson method The diffusion term is
replaced by the average of the central differences at time levels n
and n+1 30 Slide 31 The left side of the equation is a central
difference of step Which is of order The equation would be of the
form - this is read truncation error of order - The method is
unconditionally stable 31 Slide 32 In terms of the grid point ( see
figure) the left side can be interpreted as the central difference
representation of at point A the right side is the average of the
diffusion term at the same point using the explicit method using
implicit method Adding this tow equations we obtains 32 Slide 33
The Beta Formulation A general form of the finite difference
equation - The method is unconditionally stable for Note: the
formulation is Crank-Nicolson implicit for the formulation is
conditionally stable for the formulation is FTCS explicit for 33
Slide 34 34 Example - 2 (Implicit Methods) Slide 35 35 Example - 2
(Implicit Methods) Slide 36 36 Example - 2 (Implicit Methods) Slide
37 37 Example - 2 (Implicit Methods) u 1,1 u 2,1 u 3,1 t 0 =0 t 1
=0.25 t 2 =0.5 t 3 =0.75 t 4 =1.0 x 1 =0.25 x 2 =0.5x 0 =0.0 x 3
=0.75 x 4 =1.0 0 0 0 0 0 0 0 0 0 0 Sin(0.25)Sin(0. 5)Sin(0.75) u
1,4 u 2,4 u 3,4 u 1,3 u 2,3 u 3,3 u 1,2 u 2,2 u 3,2 Slide 38 38
Example - 2 (Implicit Methods) Solution of Row 1 at t1=0.25 sec
Slide 39 39 Example - 2 (Implicit Methods) Solution of Row 2 at
t2=0.5 sec u 1,1 u 2,1 u 3,1 t 0 =0 t 1 =0.25 t 2 =0.5 t 3 =0.75 t
4 =1.0 x 1 =0.25 x 2 =0.5x 0 =0.0 x 3 =0.75 x 4 =1.0 0 0 0 0 0 0 0
0 0 0 Sin(0.25)Sin(0. 5)Sin(0.75) u 1,4 u 2,4 u 3,4 u 1,3 u 2,3 u
3,3 u 1,2 u 2,2 u 3,2 Slide 40 40 Example - 2 (Implicit Methods)
Solution of Row 3 at t3=0.75 sec Slide 41 41 Example - 2 (Implicit
Methods) Solution of Row 4 at t4=1 sec Slide 42 42 Remarks The
Explicit Method: One needs to select small time step to ensure
stability. Computation per point is very simple but many points are
needed. Implicit Nicolson: Requires the solution of a Tri-diagonal
system. Stable (Larger time step can be used). Slide 43
Applications - 1 Various finite difference equations were used to
represent the parabolic model equation It is important to write
computer codes and analyzing the results which give additional
insights into the solution procedures are gained 43 Slide 44
Example Consider a fluid bounded by two parallel plates extended to
infinity such that no end effects are encountered. The planar walls
and the fluid are initially at rest. The lower wall is suddenly
accelerated in the x-direction the Navier-Stokes equations for this
problem may be expressed as. 44 Slide 45 - The fluid is oil with a
kinematic viscosity of - The spacing h = 40 mm. - The velocity of
the lower wall - It is required to compute the velocity profile - A
solution for the velocity is to be obtained up to 1.08 seconds 45
Slide 46 The initial and boundary conditions for this problem are
stated as Initial condition Boundary conditions Let assume various
of time step is to be used to investigate the numerical schemes and
the effect of time step on the stability. 46 Slide 47 47 Slide 48
Solve the above example with the following methods 1)The FTCS
explicit method with I. II. 2)The DuFort-Frankel explicit method
with I. II. 3)The Laasonen implicit method with I. II. 4)The
Crank-Nicolson method with I. II. 48 Slide 49 Solution Case 1.I the
FTCS explicit method is to be used the stability require the term
is known as diffusion number for this case the diffusion number is
therefore the stability condition is satisfied 49 Slide 50 50 Slide
51 51 Slide 52 Case 1.II the FTCS explicit method is to be used for
this case the diffusion number is therefore the stability condition
is not satisfied 52 Slide 53 53 Slide 54 Analysis In the earlier
section, various finite difference formulation were applied to the
PDE and ODE The effect of the stability imposed by the diffusion
number on the FTCS explicit method This method the step sizes is
limited due to the stability The implicit methods are
unconditionally stable and allow larger time steps. But the
accuracy requirement limits the use of large time step (increase
the time step will increase the truncation errors) 54 Slide 55
Analysis For the simple problem, an analytical solution may be
obtained This analytical result is used for: Code validation.
Comparison of various methods Study the effect of step size on the
accuracy of the solution An error term is defined as: A comparison
of various methods. 55 Slide 56 Analysis Comparison of error
distribution for various schemes at t=0.18 using the following
error term: 56 Slide 57 Analysis Comparison of error distribution
for various schemes at t=1.08 using the following error term: 57
Slide 58 Analysis Comparison of error distribution for different
time steps by using Lassonen method at t = 1.0 sec. 58 Slide 59
Parabolic Equations in Two-Dimension Various finite difference
formulations of parabolic PDEs have been discussed for 1-D The
space dimension is extended to two An efficient method of solution
is presented. Consider the model equation. Where is considered to
be constant. 59 Slide 60 Parabolic Equations in Two-Dimension
Consider an explicit formulation. By using forward differencing for
the time derivative and central differencing for the space
derivatives. This is read truncation error of order The method is
stable for Define the diffusion numbers The stability requirement
is expressed as 60 Slide 61 Parabolic Equations in Two-Dimension
Consider an implicit formulation. By using backward differencing
for the time derivative and central differencing for the space
derivatives. By defining the coefficients of the unknowns as a, b,
c, d and e and the right hand side by f, the equation may be
written as 61 Slide 62 Parabolic Equations in Two-Dimension
Consider the 5 by 5 grid system as shown in the Figure There are
nine unknown at time level n+1 A total of nine simultaneous
equations must be solved. The implicit finite difference equations
for the grid system are 62 Slide 63 63 Slide 64 The known
quantities from the imposed boundary conditions have been moved to
the right side and added to the known quantities from the previous
n time level. The set of equations can be written in a matrix form
as. 64 Slide 65 The coefficient matrix is pent-diagonal. The
solution procedure for this matrix is very time consuming. One way
to overcome the time consuming is by using a splitting method. This
method is known as alternating direction implicit method ADI. The
algorithm produces two sets of tri-diagonal simultaneous equations
to be solved in sequence. The finite difference equations of the
model in ADI formulation are 65 Slide 66 - The method is of order -
The method is unconditionally stable 66 ADI method Slide 67 The
method can be written in the tri-diagonal form as: 67 Slide 68 ADI
method solution procedure Start the solution with the first
equation, the formulation is implicit in the x-direction and
explicit in the y-direction; the solution at this stage is referred
to as the x sweep. Then the solution with the second equation, the
formulation is implicit in the y-direction and explicit in the
x-direction; the solution at this stage is referred to as the y
sweep. Graphical presentation of the method is shown in the
following Figure. 68 Slide 69 69 Slide 70 70 Application - 2 Slide
71 71 Consider the 2-D heat conduction equation Where the thermal
diffusivity It is required to determine the temperature
distribution in a long bar with a rectangular cross-section. Assume
the bar is composed of chrome steel, which has cross sectional
dimensions of 3.5 ft by 3.5 ft (b=h=3.5) Initial condition t = 0
Boundary condition Slide 72 72 The rectangular plate with B.C.
Application - 2 Slide 73 Application of the ADI method: FDE for
x-sweep By defining the coefficients for the tri-diagonal system
as: Application - 2 Slide 74 74 Application of the ADI method: Note
that in order to rearrange the equations as tridiagonal system
C(I,4) must be modified at i = 2 & i = IM-1. At I = 2 At I =
IM-1 = IMM1 Since are known from the boundary conditions and
constant Application - 2 Slide 75 75 Application of the ADI method:
FDE for y-sweep By defining the coefficients for the tri-diagonal
system as: Application - 2 Slide 76 76 Application of the ADI
method: Note that in order to rearrange the equations as
tridiagonal system C(I,4) must be modified at J = 2 & J = JM-J.
At J = 2 At J = JM-1 = JMM1 Since are known from the boundary
conditions and constant Application - 2 Slide 77 77 Application of
the ADI method: Tri-diagonal matrix can be solved by using the
Gaussian elimination method According to Gaussian elimination
method, we multiply the second equation by and the first by and
then take the difference of the two to eliminate The resulting
equation is If we replace the following Application - 2 Slide 78 78
Application of the ADI method: The same process is repeated for i =
3, 4, .., n-1 Where I = 2,3,4,,n-1 and n = IM-2 or JM-2 The value
of can immediately be found by solving simultaneously the last two
equations Application - 2 Slide 79 79 Application of the ADI
method: The remaining unknowns can be calculated in a backward
order the following formula j = n-1, n-2, n-3, , 2, 1 Application -
2 Slide 80 80 Initial temperature distribution. Temperature
distribution at t = 0.1 hr. Temperature distribution at t = 0.4 hr.
See pages (83 - 84) Application - 2 Slide 81 Sections 3.8 .. 3.13
for advance courses 3.8 Approximate Factorization 3.9 Fractional
Step Methods 3.10 Extension to Three-Space Dimensions 3.11
Consistency Analysis of Finite Difference Equations 3.12
Linearization 3.13 Irregular Boundaries Slide 82 END OF Parabolic
Partial Differential Equations 82
