BHYWI-08: Semester-FahrplanVorlesungen

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Hydroinformatik II

”Prozesssimulation und Systemanalyse”

BHYWI-08-06 @ 2018

Finite-Differenzen-Verfahren I

Olaf Kolditz

*Helmholtz Centre for Environmental Research – UFZ

1Technische Universitat Dresden – TUDD

2Centre for Advanced Water Research – CAWR

18.05./01.06.2018 - Dresden

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Konzept

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Fahrplan

Vorlesung

I Grundlagen der Finite Differenzen Methode

I Approximation methods

I Finite difference method – FDM (Ch. 3)

I Taylor series expansion

I Derivatives

I Diffusion equation

I (Finite element method – FEM ⇒ Hydrosystemanalyse)

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Naherungsverfahren

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FDM Anwendungen - MODFLOW

http://water.usgs.gov/pubs/FS/FS-121-97/images/fig7.gif

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Ableitungen

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Taylor-Reihe

in time

un+1j =

∞∑m=0

∆tm

m!

[∂mu

∂tm

]nj

(1)

∆t = tn+1 − tn

in space

unj+1 =∞∑

m=0

∆xm

m!

[∂mu

∂xm

]nj

(2)

∆x = xj+1 − xj

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Trunkation

un+1j = unj + ∆t

[∂u

∂t

]nj

+∆t2

2

[∂2u

∂t2

]nj

+ 0(∆t3) (3)

unj+1 = unj + ∆x

[∂u

∂x

]nj

+∆x2

2

[∂2u

∂x2

]nj

+ 0(∆x3) (4)

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1. Ableitung

[∂u

∂t

]nj

=un+1j − unj

∆t− ∆t

2

[∂2u

∂t2

]nj

+ 0(∆t2) (5)

[∂u

∂x

]nj

=unj+1 − unj

∆x− ∆x

2

[∂2u

∂x2

]nj

+ 0(∆x2) (6)

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Differenzen-Schemata

Forward difference approximation[∂u

∂x

]nj

=unj+1 − unj

∆x+ 0(∆x) (7)

Backward difference approximation[∂u

∂x

]nj

=unj − unj−1

∆x+ 0(∆x) (8)

Central difference approximation[∂u

∂x

]nj

=unj+1 − unj−1

2∆x+ 0(∆x2) (9)

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Zentrale Differenzen

unj+1 = unj + ∆x

[∂u

∂x

]nj

+∆x2

2

[∂2u

∂x2

]nj

+ 0(∆x3)

unj−1 = unj −∆x

[∂u

∂x

]nj

+∆x2

2

[∂2u

∂x2

]nj

− 0(∆x3) (10)

Central difference approximation[∂u

∂x

]nj

=unj+1 − unj−1

2∆x+ 0(∆x2) (11)

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Ableitungen

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2. Ableitung

[∂2u

∂x2

]nj

≈ 1

∆x

([∂u

∂x

]nj+1

−[∂u

∂x

]nj

)

≈unj+1 − 2unj + unj−1

∆x2(12)

[∂2u

∂x2

]nj

=unj+1 − 2unj + unj−1

∆x2+

∆x2

12

[∂4u

∂x4

]nj

+ ... (13)

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Ubersicht Differenzenverfahren

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Diffusionsgleichung

∂u

∂t− α∂

2u

∂x2= 0 (14)

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Analytical solution for diffusion equation (Skript 5.2.2)

I Diffusion equation

∂u

∂t− α∂

2u

∂x2= 0 (15)

I Analytical solution

u = sin(πx)e−αt2

(16)

I K: validity

⇒ Ubung

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Ubersicht Differenzenverfahren

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Explizite FDM - FTCS Verfahren (Skript 3.2.2/4.1)

I PDE for diffusion processes

∂u

∂t− α∂

2u

∂x2= 0 (17)

I forward time / centered space[∂u

∂t

]nj

≈un+1j − unj

∆t

[∂2u

∂x2

]nj

≈unj−1 − 2unj + unj+1

∆x2(18)

I substitute

un+1j − unj

∆t− α

unj−1 − 2unj + unj+1

∆x2= 0 (19)

I FTCS scheme for diffusion equations

un+1j = unj +

α∆t

∆x2(unj−1 − 2unj + unj+1) , Ne =

α∆t

∆x2(20)

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Eigenschaften numerischer Verfahren

Analysis of approximation schemes consists of three steps:

I Develop the algebraic scheme,

I Check consistency of the algebraic approximate equation,

I Investigate stability behavior of the scheme.

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Eigenschaften numerischer Verfahren

Analysis of approximation schemes consists of three steps:

I Develop the algebraic scheme,

un+1j = unj +

α∆t

∆x2(unj−1 − 2unj + unj+1) (21)

I Check consistency of the algebraic approximate equation,

lim∆t,∆x→0

|L(unj )− L(u[tn, xj ])| = 0 (22)

I Investigate stability behavior of the scheme.

Ne = α∆t∆x2≤1/2

(23)

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Losung des FTCS Schemas

Algebraische Schema

un+1j = unj +

α∆t

∆x2(unj−1 − 2unj + unj+1) (24)

Resultierendes Gleichungssystem

un+1 = Aun , n = 0, 1, 2, ... (25)

A =

1− 2. . .

. . .. . .

. . .. . .

. . .. . .

1− 2

, un =

un2un3...

unnp−2

unnp−1

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Explizite und implizite Differenzenverfahren

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Implizites Differenzenverfahren: Next Lecture

Algebraische Schema:[∂2u

∂x2

]n+1

j

≈un+1j−1 − 2un+1

j + un+1j+1

∆x2(26)

un+1j − unj

∆t− α

un+1j−1 − 2un+1

j + un+1j+1

∆x2= 0 (27)

α∆t

∆x2(−un+1

j−1 + 2un+1j − un+1

j+1 ) + un+1j = unj (28)

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BHYWI-08: Semester-FahrplanVorlesungen

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