Lecture 11: Unsteady Conduction
Error Analysis
1
Last Time…

Looked at unsteady 2D conduction problems

Discretization schemes
» Explicit
» Implicit

Considered the properties of this scheme
2
This Time…

Complete unsteady scheme discussion
» Properties of Crank-Nicholson scheme

Truncation error analysis
3
Unsteady Diffusion

Governing equation:

Integrate over control volume and time step
4
Discrete Equation Set

Dropping superscript (1) for
compactness

Notice old values in discrete
equation

To make sense of this
equation, we will look at
particular values of time
interpolation factor f (f = 0,
1, 0.5)
5
Crank-Nicholson Scheme

Crank-Nicholson scheme uses f=0.5
1
Linear profile
assumption
across time
step

0
t0
t1
t
6
Crank-Nicholson: Discrete Equations

Linear equation
set at current
time level
7
Properties of Crank-Nicholson Scheme

Linear algebraic set at current time level – need linear
solver

When steady state is reached, P= 0P. In this limit,
steady discrete equations are recovered.
» Steady state does not depend on history of time
stepping
» Will get the same answer in steady state by time
marching as solving the original steady state
equations directly
Will show later that truncation error is O (t2)

8
Properties (cont’d)

What if

Crank-Nicholson scheme can be shown to be
unconditionally stable. But if time step is too large, we
can obtain oscillatory solutions
9
Truncation Error: Spatial Approximations

Face mean value of Je represented by face centroid
value

Source term represented by centroid value:
S V   SC  SPP  V


Gradient at face represented by linear variation
between cell centroid:
   E  P
  
 xe
 x e
What is the truncation error in these approximations?
10
Mean Value Approximation

Consider 1-D approximation and uniform grid.

What is the error in representing the mean value over
a face (or a volume) by its centroid value?

Expand in Taylor series about P
11
Mean Value Approximation (cont’d)

Integrate over control volume:
12
Mean Value Approximation (cont’d)

Complete integration to find:

Rearranging:

Thus   P is a second-order accurate
representation
13
Gradient Approximation
  
 
 x e

To find
expand about ‘e’ face

Subtract to find:

Linear profile assumption => second-order
approximation
14
Temporal Truncation Error: Implicit
Scheme

Cell centroid value () = ()P at both time levels
» Same as the mean value approximation
» Second-order approximation

Value of P1 prevails over time step

Source term at (1) prevails over time step

What is the truncation error of these two
approximations?
15
Truncation Error in Implicit Scheme
(cont’d)

Consider a variable S(t) which we want to integrate
over the time step:

Expand in Taylor series about new time level (1):
16
Truncation Error in Implicit Scheme
(cont’d)

Integrate over time step:

Thus, representing the mean value over the time step
by S  S 1 is a first-order accurate approximation
17
Closure
In this lecture, we:

Completed the examination of unsteady schemes:
» Crank-Nicholson

Looked at truncation error in various spatial and
temporal approximations
18