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Lecture 11: Unsteady ConductionError Analysis

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Last Time…

Looked at unsteady 2D conduction problems

Discretization schemes» Explicit» Implicit

Considered the properties of this scheme

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This Time…

Complete unsteady scheme discussion» Properties of Crank-Nicholson scheme

Truncation error analysis

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Unsteady Diffusion

Governing equation:

Integrate over control volume and time step

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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)

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Crank-Nicholson Scheme

Crank-Nicholson scheme uses f=0.5

tt1t0

0

1Linear profile assumption across time step

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Crank-Nicholson: Discrete Equations

Linear equation

set at current

time level

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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)

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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

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Truncation Error: Spatial Approximations

Face mean value of Je represented by face centroid

value Source term represented by centroid value:

Gradient at face represented by linear variation between cell centroid:

What is the truncation error in these approximations?

C P PS V S S V

E P

e ex x

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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

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Mean Value Approximation (cont’d)

Integrate over control volume:

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Mean Value Approximation (cont’d)

Complete integration to find:

Rearranging:

Thus is a second-order accurate

representationP

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Gradient Approximation

To find expand about ‘e’ face

Subtract to find:

Linear profile assumption => second-order approximation

ex

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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?

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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):

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Truncation Error in Implicit Scheme (cont’d)

Integrate over time step:

Thus, representing the mean value over the time step

by is a first-order accurate approximation1S S

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Closure

In this lecture, we:

Completed the examination of unsteady schemes:» Crank-Nicholson

Looked at truncation error in various spatial and

temporal approximations