Direct Numerical Simulations of Multiphase Flows

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Numerical Methods for Parabolic Equations
Instructor Hong G. Im University of Michigan
Fall 2005
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Outline
Solution Methods for Parabolic Equations

• One-Dimensional Problems
• Explicit, implicit, Crank-Nicolson
• Accuracy, stability
• Various schemes
• Keller Box method and block tridiagonal system
• Multi-Dimensional Problems
• Alternating Direction Implicit (ADI)
• Approximate Factorization of Crank-Nicolson

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Numerical Methods for One-Dimensional Heat
Equations
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In this lecture, we consider a model equation
which is a parabolic equation requiring
Initial Condition
and
Boundary Condition (Dirichlet)
or
Boundary Condition (Neumann)
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Explicit Method FTCS - 1
Explicit FTCS
n1
n
j?1 j j1
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Explicit Method FTCS - 2
Modified Equation
where
- Accuracy
- If then
- No odd derivatives dissipative
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Explicit Method FTCS - 3
Stability von Neumann Analysis
(Recall Lecture 2, p. 60)
Fourier Condition
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Explicit Method FTCS - 4
Domain of Dependence for Explicit Scheme
t
P
Boundary effect is not felt at P for many time
steps This may result in unphysical
solution behavior
?t
h
BC
BC
x
Initial Data
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Implicit Method - 1
Implicit Method Laasonen (1949)
Tri-diagonal matrix system
n1
n
j?1 j j1
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Implicit Method - 2
Modified Equation
- The sign suggests that implicit method may
be less accurate than a carefully implemented
explicit method.
Amplification Factor (von Neumann analysis)
Unconditionally stable
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Crank-Nicolson - 1
Crank-Nicolson Method (1947)
Tri-diagonal matrix system
n1
n
j?1 j j1
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Crank-Nicolson - 2
Modified Equation
- Second-order accuracy
Amplification Factor (von Neumann analysis)
Unconditionally stable
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Combined Method A - 1
Generalization
n1
n
j?1 j j1
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Combined Method A - 2
Modified Equation
Other Special Cases (for further accuracy
improvement)
(a) If
(b) If and
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Combined Method A - 3
Stability Property
? unconditionally stable
? stable only if
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Combined Method B - 1
Generalized Three-Time-Level Implicit
Scheme Richtmyer and Morton (1967)
j?1 j j1
n1
n
n?1
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Combined Method B - 2
Modified Equation
Special Cases
(a) If
(b) If
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Richardson Method
Richardson Method A Case of Failure
Similar to Leapfrog
but unconditionally unstable!
n1
n
n?1
j?1 j j1
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DuFort-Frankel - 1
The Richardson method can be made stable by
splitting by time average
n1
n
n?1
j?1 j j1
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DuFort-Frankel - 2
Modified Equation (Recall Homework 1)
Conditionally consistent
Amplification factor
Unconditionally stable
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Parabolic Equation - Summary
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Parabolic Equation - Summary
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Keller Box - 1
The Keller Box Method (1970)
- Implicit with
Basic Concept
Define
yielding
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Keller Box - 2
In discretized form
n1
?tn1
hj
n
j?1 j
n1
where
n
j?1 j
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Keller Box - 3
Substituting
or
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Keller Box - 4
Adding boundary conditions, e.g.
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Keller Box - 5
In Matrix Form
Block Tridiagonal Matrix ? Appendix, Linpack
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Keller Box - 6
Modified Keller Box Method Express in terms
of
Starting with original Keller box method
(a)
(b)
Eliminating from (b) using (a)
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Keller Box - 7
Further elimination of yields
(Tannehill, p. 136)
Tridiagonal Matrix ? Thomas Algorithm
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Keller Box - 8
Notes on Keller Box Method

• Second-order accurate in time and space
• Accuracy is preserved for nonuniform grids
• More operations per timestep compared to
• Crank-Nicolson

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Numerical Methods for Multi-Dimensional Heat
Equations
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Explicit Method - 1
Consider a 2-D heat equation
Applying forward Euler scheme
If
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Explicit Method - 2
In matrix form
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Explicit Method - 3
Von Neumann Analysis
Worst case
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Explicit Method - 4
Stability Condition for Heat Equation
(2-D)
(3-D)
(1-D)
Explicit method for multi-dimensional heat
equation is not desirable.
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Crank-Nicolson
Crank-Nicolson Method for 2-D Heat Equation
If
Too expensive!!
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ADI - 1
A Clever Remedy Alternating Direction Implicit
(ADI)
Fractional Step
Step 1
Step 2
Combining the two becomes equivalent to
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ADI - 2
Computational Molecules for ADI Method
n1
n1/2
n
j1
j
i?1
i
j?1
i1
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ADI - 3
ADI Method is accurate
Stability Analysis
Similarly,
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ADI - 4
Combining
Unconditionally stable!
Unfortunately, a 3-D version does not have the
same desirable stability properties. ?
Conditionally stable and
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ADI - 5
Advantages

• Stability limits of 1-D case apply.
• Different ?t can be used in x and y directions.
• 1. Cannot be directly extended to 3-D problems.
• ? Approximate factorization

Limitations
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Approximate Factorization - 1
Define
The Crank-Nicolson for heat equation becomes
which can be rewritten as
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Approximate Factorization - 2
Factoring each side
or
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Approximate Factorization - 3
Final factored form of the discrete equation
at an accuracy of
Note that the ADI method can be written as
which is equivalent to factorized Crank-Nicolson
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Approximate Factorization - 4
Two-step algorithm
can be written into two steps
(Step 1)
(Step 2)
each of which can be solved by TDMA (Thomas
algorithm).
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Approximate Factorization - 5
Note Boundary condition for is needed at
This can be determined from
For example, at
Similarly, at
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Approximate Factorization - 6
Generalized 3-D Algorithm unconditionally
stable,
where
(Step 1)
(Step 2)
(Step 3)