finite volume numerical solutions to the shallow water ... · to implement fvm to solve hyperbolic...

Finite Volume Numerical Solutions tothe Shallow Water wave Equations

Dr Sudi Mungkasi

Department of MathematicsSanata Dharma University, Yogyakarta, Indonesia

Email: [email protected]

Presented at:The Australian National University, Canberra, Australia

25 June 2019

Dr Sudi Mungkasi (ANU Visitor) Finite volume methods 1 / 56

Contents

In this talk, we present

Motivation and aims

Conservation laws and FVM

FVM for SWE

Adaptive mesh FVM for SWE


USA: Johnstown dam break

More than 2000 people died: USA, 31 May 1889

Ref: http://projectdisaster.com/?m=200805


France: Malpasset dam break

More than 400 people died: France, 2 Dec 1959

Ref: http://www.ncche.olemiss.edu/software/ccheflood


Indonesia: Aceh tsunami 2004

More than 166,000 people died: Indonesia, 26 Dec 2004

Ref: http://medlem.spray.se/meunasah/


Japan: Tohoku tsunami 2011

At least 10,000 people dead: Japan, 11 Mar 2011

Ref: http://news.nationalgeographic.com/news/2011/03/


Aims

Our aims are

to implement FVM to solve hyperbolic PDE

to obtain numerical solutions to SWE

to see some open problems in plasma physics that may be solvedusing FVM

To anticipate floods due to dam break and tsunami:

early warning system

modelling of water flows

simulation of water flows, such as using ANUGA software.


Contents

Contents

Motivation and aims


FVM for SWE



Conservative form

In one space dimension, conservation laws have the form:

qt + f (q)x = 0. (1)

When there is a source term s, it is usually called balance laws and has theform:

qt + f (q)x = s. (2)


Specific examples of conservation laws:

a). advection equation: qt + cqx = 0.

b). Burgers’ equation: ut + (12u2)x = 0.

c). shallow Water wave Equations (SWE) :

∂Q

∂t+

∂F(Q)

∂x+

∂G(Q)

∂y= 0 (3)

where

Q =

w

uh

vh

, F(Q) =

uh

u2h + 12gh

2

uvh

, G(Q) =

vh

uvh

v2h + 12gh

2

(4)


Wave equation

Recall the second order scalar wave equation

utt = c2uxx , −∞ < x < ∞. (5)

It can be written as a first order system of conservation laws as follows.Introducing

v = ux , w = ut (6)

we then have vt = wx and wt = c2vx .Therefore we obtain the system

vt + (−w)x = 0, (7)

wt + (−c2v)x = 0. (8)

That is,qt + f (q)x = 0 (9)

in which q = [v , w ]T and f (q) = [−w , − c2v ]T .Dr Sudi Mungkasi (ANU Visitor) Finite volume methods 11 / 56

Heat equation

The one-dimensional heat equation is

qt = cqxx (10)

where c is positive constant, which is the diffusion coefficient. The heatequation can be rewritten into a conservative form

qt + (−cqx)x = 0 . (11)


Methods to solve conservation laws

Finite Difference Method works from the differential form of conservationlaws

qt + f (q)x = 0. (12)

Finite Volume Method works from integral forms of conservation laws

d

dt

∫ x2

x1

q(x , t) dx = f (q(x1, t))− f (q(x2, t)). (13)

or∫ x2

x1

q(x , t2) dx =

∫ x2

x1

q(x , t1) dx

+

∫ t2

t1

f (q(x1, t))dt −

∫ t2

t1

f (q(x2, t))dt. (14)


Finite Volume Methods

We use the following notations

Ci = (xi− 12, xi+ 1

2) (15)

Qni ≈

1

∆x

∫ xi+1

2

xi− 1

2

q(x , tn)dx ≡1

∆x

∫

Ci

q(x , tn)dx (16)

F ni+ 1

2

≈1

∆t

∫ tn+1

tn

f [q(xi+ 12, t)]dt (17)


Finite Volume Methods

Then the integral form of conservation laws

∫ x2

x1

q(x , t2) dx =

∫ x2

x1

q(x , t1) dx

+

∫ t2

t1

f (q(x1, t))dt −

∫ t2

t1

f (q(x2, t))dt. (18)

can be written as a fully-discrete Finite Volume Scheme

Qn+1i = Qn

i −∆t

∆x(F n

i+ 12

− F ni− 1

2

). (19)

Here F ni+ 1

2

for all i are the fluxes.


Contents

Contents

Motivation and aims


FVM for SWE



1D SWE

The 1D SWE isqt + f(q)x = s ,

where

q =

[

h

hu

]

, f =

[

hu

hu2 + 12gh

2

]

, s =

[

0−ghzx

]

.


FVM for 1D SWE

Recall that forqt + f (q)x = 0, (20)

we obtain a semi-discrete FVM

∆xid

dtQi + F n

i+ 12

− F ni− 1

2

= 0 (21)

or a fully-discrete FVM

Qn+1i = Qn

i −∆t

∆xi(F n

i+ 12

− F ni− 1

2

). (22)


FVM for 1D SWE

Then forqt + f (q)x = s, (23)

we obtain a semi-discrete FVM

∆xid

dtQi + F n

i+ 12

− F ni− 1

2

= Si (24)

or a fully-discrete FVM

Qn+1i = Qn

i −∆t

∆xi(F n

i+ 12

− F ni− 1

2

) +1

∆xiSi . (25)


Simulation example 1: dam-break problem

The following is by a 2nd order method.

! " # $ % & % # ' ( ) *+ ,Dr Sudi Mungkasi (ANU Visitor) Finite volume methods 20 / 56

Simulation example 2: perturbation of a lake at rest

The following is by a 2nd order FVM.- . / 0 12 1 3 4 5 1 6 / . 7 8 9: ; : : ; < = ; : = ; < > ; :: ; :: ; >: ; ?: ; @: ; A= ; :B C DEF: ; : : ; < = ; : = ; < > ; :G = ; :G : ; <: ; :: ; <= ; :H IJFKC LJ: ; : : ; < = ; : = ; < > ; :M N O P Q P N RG = ; :G : ; <: ; :: ; <= ; :S FT IUVC W


2D SWE

The two-dimensional shallow water equations are

∂Q

∂t+

∂F(Q)

∂x+

∂G(Q)

∂y= 0 (26)

where

Q =

w

uh

vh

, F(Q) =

uh

u2h + 12gh

2

uvh

, G(Q) =

vh

uvh

v2h + 12gh

2

(27)


General forms

Conservation laws in one-dimensional space have the form

qt + f (q)x = 0. (28)

Conservation laws in two-dimensional space have the form

qt + f (q)x + g(q)y = 0. (29)

Here, we focus on two-dimensional space domain. We will discretise thedomain into polygonal cells: usually either rectangular or triangular grids.


Two approaches of FVM for 2D SWE

Two popular approaches to solve the 2D SWE using FVM:

Rectangular grids

Triangular grids


1. Rectangular grids

Discretisation of the spatial domain

The value Qnij represents a cell average over the (i , j) grid cell at time tn,

Qnij ≈

1

∆x∆y

∫ y+ 12

y− 12

∫ x+ 12

x− 12

q(x , y , tn) dx dy . (30)



Recall qt + f (q)x + g(q)y = 0. Let Cij = [xi− 12, xi+ 1

2]× [yj− 1

2, yj+ 1

2],

∆x = xi+ 12− xi+ 1

2and ∆y = yj+ 1

2, yj− 1

2. Then we have

ddt

∫ ∫

Cij

q(x , y , t) dx dy =

−

∫ yj+1

2

yj− 1

2

f (q(xi+ 12, y , t)) dy +

∫ yj+1

2

yj− 1

2

f (q(xi− 12, y , t)) dy

−

∫ xi+1

2

xi− 1

2

f (q(x , yi+ 12, t)) dx +

∫ xi+1

2

xi− 1

2

f (q(x , yi− 12, t)) dx (31)



Integrating this expression from tn to tn+1 and dividing it by the cell area∆x∆y , we obtain a fully-discrete FVM for Conservation laws intwo-dimensional space have the form

qt + f (q)x + g(q)y = 0 (32)

is

Qn+1ij = Qn

ij −∆t

∆x

[

F ni+ 1

2,j− F n

i− 12,j

]

−∆t

∆y

[

Gni ,j+ 1

2

− Gni ,j− 1

2

]

. (33)

Here

F ni− 1

2,j≈

1

∆t∆y

∫ tn+1

tn

∫ yj+1

2

yj− 1

2

f (q(xi− 12, y , t)) dy dt, (34)

Gni ,j− 1

2

≈1

∆t∆x

∫ tn+1

tn

∫ xi+1

2

xi− 1

2

g(q(x , yj− 12, t)) dx dt. (35)


2. Triangular grids

Suppose that we discretise the domain into a finite number of triangles.

Discretisation of the spatial domain


FVM on arbitrary polygonal grids

The conservation laws with source term can be written as

∂

∂t

∫

Ωq dΩ+

∮

ΓT−1f(Tq) dΓ =

∫

Γs dΩ . (36)

The discrete version of it is

dqidt

+1

Ai

∑

j∈N (i)

Hij lij = si . (37)

Here T is a transformation, which transform a vector from globalcoordinates into local coordinates. The local coordinates are the edges ofthe polygonal cell and their orthogonal direction.


Algorithm for FVM on arbitrary polygonal grids

Recall the discrete version of the FVM

dqidt

+1

Ai

∑

j∈N (i)

Hij lij = si . (38)

1 For each interface (i , j) , transform the quantity qi and qj in theglobal coordinate system (x , y) into the quantity qi and qj in thelocal coordinate system system (x , y) .

2 Compute the flux f at the interface (i , j) corresponding to

qt + f(q)x = s (39)

3 Transform the flux f back to the global coordinate system (x , y).

4 Finally, solve (38) where N (i) = 0, 1, 2, if triangular grids areconsidered, for qi .


ANUGA Software

Some descriptions:

1 ANUGA is an open source (free software) developed by AustralianNational University (ANU) and Geoscience Australia (GA).

2 The mathematical background underlying the software is the finitevolume method.

3 Triangular computational grids are used.

4 The interface of this software is in Python language, but thecomputationally expensive parts are written in C language.

5 A thorough description of this software is available athttp://anuga.anu.edu.au/.


Planar dam break at 0 s

Figure : Initial condition for the planar dam break problem. ANUGA software wasused in this simulation.


Planar dam break at 0.5 s

Figure : Water flows 0.5 second after the planar dam is broken. ANUGA softwarewas used in this simulation.



Figure : Water flows 1.0 second after the planar dam is broken. ANUGA softwarewas used in this simulation.



Figure : Water flows 1.5 seconds after the planar dam is broken. ANUGAsoftware was used in this simulation.


Circular dam break at 0 s

Figure : Initial condition for the circular dam break problem. ANUGA softwarewas used in this simulation.


Circular dam break at 0.5 s

Figure : Water flows 0.5 second after the circular dam is broken. ANUGAsoftware was used in this simulation.



Figure : Water flows 1.0 second after the circular dam is broken. ANUGAsoftware was used in this simulation.



Figure : Water flows 1.5 seconds after the circular dam is broken. ANUGAsoftware was used in this simulation.


Contents

Contents

Motivation and aims


FVM for SWE



Weak local residuals (WLR)

Weak local residual (WLR) can be used as a smoothness indicator ofnumerical solutions.Consider the scalar balance law with an initial condition

qt + f (q)x = s , −∞ < x < ∞ ,

q(x , t) = q0(x) , t = 0 .

The weak form of the initial value problem is∫

∞

0

∫∞

−∞

[q(x , t)Tt(x , t) + f (q(x , t))Tx(x , t) + s(x , t)T (x , t)] dx dt

+

∫∞

−∞

q0(x)T (x , 0) dx = 0,

where T (x , t) is an arbitrary test function.



We can choose the localised linear B-splines as the test functions

Tn−1/2j+1/2 (x , t) := Bj+1/2(x)B

n−1/2(t) , where

Bj+1/2(x) =

x−xj−1/2

∆xif xj−1/2 ≤ x ≤ xj+1/2 ,

xj+3/2−x

∆xif xj+1/2 ≤ x ≤ xj+3/2 ,

0 otherwise ,

and Bn−1/2(t) is defined similarly.This results in the WLR

En−1/2j+1/2 = −

∫ tn+1/2

tn−3/2

∫ xj+3/2

xj−1/2

[

q∆(x , t)[

Tn−1/2j+1/2

]

t

+ f (q∆(x , t))[

Tn−1/2j+1/2

]

xs∆(x , t)T

n−1/2j+1/2

]

dx dt.



After a straightforward computation, we obtain the WLR formulation

En−1/2j+1/2 =

∆x

2

[

qnj − qn−1j + qnj+1 − qn−1

j+1

]

+∆t

2

[

f(

qn−1j+1

)

− f(

qn−1j

)

+ f(

qnj+1

)

− f(

qnj)

]

−∆x∆t

4

[

sn−1j + snj + sn−1

j+1 + snj+1

]

. (40)

This WLR can be extended into 2D.


Adaptive Methods in 1D

0 5 10 15 20 250.00.20.40.60.81.01.2

Stage

StageBed

0 5 10 15 20 250.000

0.010

0.020

CK

0 5 10 15 20 25Position

−0.06−0.04−0.020.000.020.040.06

Cell level

WLR in adaptive methods in 1D.



0 5 10 15 20 250.00.20.40.60.81.01.2

Stage

StageBed

0 5 10 15 20 25

0.0000.0100.020

CK

0 5 10 15 20 25Position

0

2

4

6

8

10

Cell level




0 5 10 15 20 250.00.20.40.60.81.01.2

Stage

StageBed

0 5 10 15 20 25

0.000

0.010

0.020

CK

0 5 10 15 20 25Position

0

2

4

6

8

10

Cell level

WLR in adaptive methods in 1D. (154 cells)


Standard Methods in 1D

0 5 10 15 20 250.00.20.40.60.81.01.2

Stage

StageBed

0 5 10 15 20 25

0.000

0.010

0.020

CK

0 5 10 15 20 25Position

−1.0

−0.5

0.0

0.5

1.0

Cell level

Standard (non-adaptive) methods in 1D using 308 cells.



Adaptive methods in 2D.



−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

xy



Contents

Contents

Motivation and aims


FVM for SWE



Conclusions

In this talk, we have presented


FVM for SWE


Other topics that could have been covered

Staggered grid FVM for SWE

Parallel computations of FVM for SWE

Application of FVM in plasma physics

Are there open problems in plasma physics that may be solved using FVM?


Main references

R. J. LeVeque. Finite volume methods for hyperbolic problems,Cambridge University Press, Cambridge, 2002.

S. Mungkasi. A study of well-balanced finite volume methods andrefinement indicators for the shallow water equations, PhD thesis,Australian National University, Canberra, 2012.


finite volume numerical solutions to the shallow water ... · to implement fvm to solve hyperbolic...

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