Å.Ødegård » Fully Implicit Methods for Systems of PDEs
Fully Implicit Methods for
Systems of PDEs
1/19
Outline
Overview, Goal
Operator splitting based methods
Example with problem
Overview Fully Implicit method
Nonlinear problem
Examples
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
2/19
Overview and Goal
In an earlier talk, Tom discussed implementations for
Systems of PDEs solved by splitting-techniques.
We focus on Implicit methods for Systems of PDEs.
Jacobi or Gauss–Seidel based solution strategies
may be either slow or unstable, or both.
Main goal: Establish framework for implementation of
fully implicit solvers.
Retain the flexibility and extensibility of the operator
splitting approach.
Talk based on chapter submitted to the next Diffpack
book.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
3/19
Re-cap: Operator splitting
we can formulate two different iterations:
such that
such that
Gauss–Seidel:
1. Compute
2. Compute
Jacobi:
such that
1. Compute
2. Then compute
such that
Given
Consider systems of differential equations on the form
System reduced to solving two scalar equations
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
4/19
Splitting is not safe
Consider the system:
The system may easily be reformulated as:
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
Analytical solution is:
5/19
Gauss–Seidel on ODE problem
The Gauss–Seidel method can be formulated in two ways
system (3), depending on in which order
for the
we solve the equations; either
or
These methods give the solution curves plotted in the figures on the next slide
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
6/19
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(b)
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Solution of
system (7), with
and
.
100
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(c)
Solution of
system (6), with
and
.
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Solution of
system (6), with
and
.
100
90
(a)
80
70
60
50
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20
10
0
−0.5
The solution of (3) approximated with Gauss–Seidel’s method.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
7/19
Jacobi on the ODE problem
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(d)
Stabilitycondition
fulfilled,
and
.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
0
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(e)
condition
satisfied,
and
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Stabilitynot
.
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8/19
Fully Implicit solver
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Å.Ødegård » Fully Implicit Methods for Systems of PDEs
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9/19
Lessons learned so far...
Simple splitting–techniques may lead to severe
difficulties.
The order of the equations in Gauss–Seidel may
affect the solution.
Simple splitting–techniques may converge to the
wrong solution.
Fully implicit method are more safe.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
10/19
Fully Implicit Implementation
Consider the system
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
and
where
:
and :
Finite element form of system (10) written with block
matrices:
11/19
Fully Implicit cont...
The Gauss–Seidel iteration written with block matrices:
The Gauss–Seidel method and the implicit method
only differ in how terms are arranged.
If we have stand–alone solvers for the sub problems,
these can be reused as before.
We need to rearrange the terms, by changing the
assembling of the linear system.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
12/19
Code Framework
.
Manager–class derived from
SystemFEM have pointer to each sub-solver.
which assemble the
.
Sub-solvers must implement the
method.
Sub-solvers must derive the
base-class.
Based on framework for Mixed Methods,
Implements a new
complete system.
Storage for full system.
This is not in Diffpack yet.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
13/19
Nonlinear Problems
:
The Newton–iteration for
We use the Newton–Raphson method.
and
Nonlinear system. Given to non–linear problems:
Assume that Diffpack solvers exist.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
Consider coupled problem:
14/19
Gauss–Seidel for Nonlinear Problem
Gauss–Seidel iteration:
The corresponding Newton iterations on the discrete
problem:
Contain only terms present in the sub-solvers.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
15/19
Fully Implicit solver
Apply Newton method to (15) directly:
The Jacobi matrices off–diagonal are new.
The rest of the terms are present in the
Gauss–Seidel solver.
This is different from the linear case.
Still, the resulting linear system is similar to what we
obtain in the linear case.
We can implement a fully implicit solver as we have
done in the linear case.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
16/19
Examples
on
on
on
Pipeflow problem
Here,
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
and
17/19
Pipeflow classes
gy1
Ener
gy2
Ener
um1
ment
Mo
um2
ment
Mo
gy3
Ener
oss
B
G−S
um3
ent
m
o
M
2
Boss
el2
onR
m
m
o
C
el
onR
m
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o
C
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
18/19
Pipeflow cont.
and
implement
and
as well as some book-keeping functions.
pow.law exponent:
redefine functions which use the
FiniteElement, since we are using the Mixed
framework.
iter.
time
iter.
time
Fully Implicit
4
4342.7s
3
3487.6s
Gauss–Seidel
3
7747.8s
3
5995.8s
Performance comparison.
is Newton iterations in the fully
implicit case and outer iterations in the Gauss–Seidel case.
Å.Ødegård » Fully Implicit Methods for Systems of PDEs
19/19