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Agenda:
Tobias Köppl TUM
2
1. Time integration methods
1.1. Implicit and explicit Onestep Methods
1.2. Convergence theory for Onestep Methods
2. The Heat Equation
2.1. Discretization of the Laplacian operator
2.2. Application of Time integration methods
2.3. Courant-Friedrichs-Levy condition
3. Outlook
4. Literature
Tobias Köppl TUM
3
Notation:
)],,([ 01 dTtCy y: is a differentiable function, which maps from the intervall
],[ 0 Tt into d
d : vector space of d dimensional vectors with components, which are in
: field of real numbers
and whose derivative function is continous
:yt partial derivative of y with respect to t
:y Laplacian operator of y
d
i i
i
x
yy
12
2
: Machine accuracy 1610
:)( fOg
[,0[:, fg
0, RC with )()( xfCxg Rx
1. Time integration methods
Tobias Köppl TUM
4
1.1. Implicit and explicit Onestep Methods
00 )(
),()(
xtx
xtftxdt
d
d
dd
x
xtfxt
tf
0
0
),(),(
[,[:
Goal: Find numerical approximations of functions )],,([ 01 dTtCx
which are solutions of an ODE with an initial value:
(IVP)
1. Time integration methods
1.1. Implicit and explicit Onestep Methods
5Tobias Köppl TUM
)()( txtxdt
d
1)0( x )exp()( ttx
Example (IVP):
Dahlquist‘s Testequation
Tobias Köppl TUM
6
1. Time integration methods
1.1. Implicit and explicit Onestep Methods
Construction of a time integration method
Divide the continous intervall ],[ 0 Tt by 1n discrete timepoints:
Tttt n ...10
Grid },...,{ 0 ntt on ].,[ 0 Tt
tfor all)()( txtx
}0|max{ 1 njtt jjj Stepsize of grid :
by a gridfunction Approximation of the solution at the gridpoints jtdx :
1t 2t 3t 2nt1ntt
0t 1 ntT
321 2n 1n
Tobias Köppl TUM
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1. Time integration methods
1.1. Implicit and explicit Onestep Methods
)()(...)()( 100 Txtxtxxtx n
Such algorithms are called Onestep Methods.
Additionaly: Computation of the gridfunction by recursion, should be possible.
Reason: For the computation of )( 1 jtx only )( jtx from the timepoint
before is needed.
Further possibility: Multistep Methods.
See literature for more information f.e. [DB II] Chapter 7 or [QSS] Chapter 11
1. Time integration methods
1.1. Implicit and explicit Onestep Methods
Tobias Köppl TUM
8
Two popular Onestep Methods are the explicit and the implicit Euler Method:
))(,()()( 111 jjjjj txtftxtx
explicit Euler Method:
implicit Euler Method:
00 )( xtx
))(,()()( 1 jjjjj txtftxtx
1. Time integration methods
1.1. Implicit and explicit Onestep Methods
Tobias Köppl TUM
9
Geometry of the explicit Euler Method:
The explicit Euler Method is creating a path (Euler path).
2l
1l
t
Construction of: ).( 1 jtx
t)( jtx
Compute the tangent vector t in).( jtx
Compute the intersection point between:
tt
txl
j
j
)(:1
and
01
0: 1
2jtl
First component of thisintersection point is the next value of the gridfunction ).( 1 jtx
Source: [DB II]
1. Time integration methods
Tobias Köppl TUM
10
1.2. Convergence theory for Onestep Methods
Issue: Under which conditions does a Onestep Method converge towards the exact solution of an ODE with an initial value?
Definition (local discretization error):
The local discretization error of a gridfunction fora grid on the intervall is defined as:
)(l
dx :],[ 0 Tt
)()(max)( 1)(
10
j
jj
njtxtxl
)( jx is the solution of the IVP:
)()(
),()(
jj txtx
xtftxdt
d
on ],[ 1jj tt
1. Time integration methods
1.2. Convergence theory for Onestep Methods
Tobias Köppl TUM
11
Definition:A Onestep Method is called constistent, if:
0)( l for .0
Theorem:The explicit and the implicit Euler Method are consistent.
Proof: See f. e. [DB II] Chapter 4.
1. Time integration methods
1.2. Convergence theory for Onestep Methods
Tobias Köppl TUM
12
Defintion (convergence):
A Onestep Method is called convergent towards the exact solution of an IVP, if:
0)( e .0for
Definition (global discretization error):
The global discretization error is the maximum error between the computedapproximations and the corresponding values of the exact solution
)( jtx).( jtx
)(e
)()(max)( 110
jj
njtxtxe
1. Time integration methods
1.2. Convergence theory for Onestep-Methods
Tobias Köppl TUM
13
Comparison of local and global Discretization error:
Source: [Bun]
1. Time integration methods
1.2. Convergence theory for Onestep-Methods
Tobias Köppl TUM
14
Defintion (stability):A numerical algorithm is called (numerically) stable, if for all permitted input data perturbed in the size of computational accuracy ( ) acceptable results are produced under the influence of rounding and method errors.
)(O
Maintheorem of Numerics:
A Onestep Method is convergent, if and only if it is consistent.
1. Time integration methods
1.2. Convergence theory for Onestep-Methods
Tobias Köppl TUM
15
Remark:Stability of a Onestep Method is an essential condition for getting qualitatively correct solutions, when using practical stepsizes. See numerical experiment.
Proof: See f. e. [Jun] Chapter 4.
Theorem:
The implicit Euler Method is stable for any stepsize
The explicit Euler Method is only stable for small .
.
Example:
If , the explicit Euler Method is a stable integrator for Dahlquist‘s testequation.
2
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
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Goal: Find a numerical approximation of the two dimensional Laplacian operator in order to solve the Poisson equation:
21,0
fu 0u
:f
(PE)
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
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Discretization Method: Finite Differences
Main idea: Replace differential operators by difference operators
Further discretization methods: Finite Volumes or Finite Elements
See literature for more information f.e. [QSS] or [Wa]
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
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Discretization of :
Discretizationpoints: ),( ji yxihxi and ,jhy j
hji1
,0
Uniform grid with grid parameter h
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
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),(, jiji yxuu
),(, jiji yxff
expand ihxi ),( ji yhxu in a Taylor series aroundand ),( ji yhxu
Discretization of the two dimensional Laplacian operator:
jihjijijijiji
ji uh
uuuuuu ,2
,1,11,1,,,
4
expand jhy j ),( hyxu ji in a Taylor series aroundand ),( hyxu ji
Discretization error: )( 4hO
),( ji yx is an interior point of the unit square
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
21
jijijijijiji f
h
uuuuu,2
,1,11,1,,4
hji1
,0
In order to get a numerical solution of the (PE) we have to solve the following System of linear equations:
Matrix- Vectornotation:
fAu (SLSE)
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
22
Structure of the Matrix A:A ist a sparse Matrix
Use fast iterative solverslike Jacobi Method or GaußSeidel Methodfor the solution of (SLSE)
See f.e. [El]
Source: [Hu]
2. The Heat Equation
2.1. Discretization of the Laplacian operator
Tobias Köppl TUM
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22
1,0
)sin()sin(2
yxu
0u
Example:
Solution: See numerical example
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
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21,00
uut0u
2)1,0(:g )(),0( xgxu
Goal: Find numerical approximations of functions u which are solutions of the homogenous Heat Equation:
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
25
Solution of the Heat Equation depends on time and space.
Discretization of time andspace is necessary
Both time and space can be discretized by an uniform grid
Source: [SK]
h
h
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
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Discretization of :
Discretizationpoints: ),( ji yxihxi and ,jhy j
hji1
,0
Uniform grid (2D) with grid parameter h
Discretization of :],[ 0 Tt
Uniform grid (1D) with stepsize k and n timepoints
timepoints: kmtm
nm 0
jimjim uyxtu ,,),,(
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
27
uuuu tt 0
Construct a local IVP for every ),( ji yxh
ji1
,0
Tt
yxgyxu
yxtuyxtudt
d
jiji
jiji
,0
),(),,0(
),,(),,(
(IVP(ij))
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
28
Solve IVP(ij) with the implicit Euler Method:
jimjimjim
jiji
ukuu
yxgu
,,1,,,,1
,,0 ),(
Use the discretization of the two dimensional Laplacian Operator:
2
,1,1,1,11,,11,,1,,1,,1
4
h
uuuuuu jimjimjimjimjim
jim
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
29
jimjimjimjimjimjim uk
huuuuu
k
h,,
2
,1,1,1,11,,11,,1,,1
2
4
We have to solve the following system of linear equations for every timestep:
nm 0h
ji1
,0
Use again fast iterative solvers like Jacobi Method or Gauß Seidel Method for the solution of (SLSE II)
(SLSE II)
2. The Heat Equation
2.2. Application of Time integration methods
Tobias Köppl TUM
30
21,00
uut0u 1),0( xu
25.0,0t
Numerical Example: (Cooling of a heated plate)
2. The Heat Equation
2.3. Courant-Friedrichs-Levy condition
Tobias Köppl TUM
31
(IVP(ij)) can be solved by the explicit Euler Method, too.
Chapter 1.2.: explicit Euler Method is only stable for a small stepsize k
Maintheorem of Numerics: Stability is an essential condition for getting qualitatively correct solutions, when using practical stepsizes.
2. The Heat Equation
2.3. Courant-Friedrichs-Levy condition
Tobias Köppl TUM
32
h
h
Theorem (CFL condition):
The explicit Euler Method is astable solver for the Heat Equation if:
142
h
k
Proof: See [CFL]
Source: [SK]
2. The Heat Equation
2.3. Courant-Friedrichs-Levy condition
Tobias Köppl TUM
33
21,00
uut0u 1),0( xu
25.0,0t
Numerical Example: (Cooling of a heated plate,solved with the explicit Euler Method and a discretizationwhich is not conform with the CFL condition)
3. Outlook
Tobias Köppl TUM
Let x(t) be the solution of an (IVP):
Adaptive discretization algorithms, with a good error measurement are required
34
Source: [DH I]
x(t)
x(t)
t
An uniform discretization of the time axis, would lead to a Onestep Methodwith slow convergence
3. Outlook
Tobias Köppl TUM
35
Further topics:
Optimization of the algorithms, solving the sparse linear equation systems,with respect to storage and number of floating point operations
Traversing a certain grid efficiently (peano curves)
Preconditioning of the Matrix representing the mentioned SLSE
4. Literature:
Tobias Köppl TUM
36
[DH I]: Numerische Mathematik I, Deuflhard/Hohmann, 2002, 3. edition[DB II]: Numerische Mathematik II, Deuflhard/Bornemann, 2002, 2. edition[Jun]: Lecture on Numerische Mathematik, Junge, 2007[SK]: Numerische Mathematik, Schwarz/Köckler, 2006, 6. edition[QSS]: Numerische Mathematik 2, Quateroni/Sacco/Saleri, 2002, 2. edition[Wa]: Lecture on Finite Elements, Wall, 2007, 2. edition[Hu]: Numerics for computer sience students, Huckle/Schneider, 2002, 3. edition[Bun]: Lecture on numerical programming, Bungartz, 2007[El]: Finite Elements and Fast Iterative Solvers, Elman/Silvester/Wathen, 2005, 2. ed.[Fo I]: Analysis I, Forster, 1976, 6. edition[Fo II]: Analysis II, Forster, 1976, 6. edition[Fei]: Introduction into the theory of partial differential equations, 2000 [CFL]: Über die partiellen Differentialgleichungen der mathematischen Physik, Courant, Friedrichs, Levy, 1928
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