Technische Universität MünchenDepartment of Informatics V
29th April, 2013
A. Breuer, S. Rettenberger, M. Bader
Session 2: Bathymetry, Dimensional Splitting, SWE
Lab: Scientific Computing - Tsunami-Simulation
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Outline
• Presentations: Assignment 1• Numerics
– Bathymetry– Dimensional Splitting
• Preparation: Assignment 2• SWE
2
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation 3
Schedule (big meetings)
Date Schedule15.04.2013 Kickoff29.04.2013 Presentation 113.05.2013 Presentation 227.05.2013 Presentation 310.06.2013 Presentation 424.06.2013 Report: Project phase01.07.2013 Presentation Project
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Presentations: Assignment 1
4
Prof. Dr. M. BaderDipl.-Math. A. BreuerS. Rettenberger, M. Sc.
TUM, SCCS (I5)Bachelor-Praktikum:Tsunami-Simulation April 15, 2013
In this assignment we are going to implement and test the most basic functionality of ourlab-course: The f-wave solver for the one-dimensional shallow water equations. The shallowwater equations are a system of nonlinear hyperbolic conservations laws with an optionalsource term:
h
hu
�
t
+
hu
hu
2 + 1
2
gh
2
�
x
= S(x, t). (1)
The quantities q = [h,hu]T are defined by h(x, t), the space-time dependent height of thewater column and hu(x, t), the space-time dependent momentum in spatial x-direction (uis the particle velocity of the water column). g the gravity constant (usually g := 9.81m/s
2)and f := [hu,hu2 + 1
2
gh
2]T the flux function. As source term S(x, t) we will consider thee↵ect of space-dependent bathymetry (topography of the ocean) only S(x) = [0,-ghB
x
]T ,embedding of additional forces, such as friction or the coriolis e↵ect is possible. Figure 1illustrates the involved quantities.
Figure 1: Sketch of quantities appearing in the one-dimensional shallow water equations.
To verify, that this fundamental functionality of our program works as expected, proper(unit-) testing is required. We will do testing by a selection of standardized tests, for whicha solution is available.
Remark As units we use meters (m) and seconds (s) for all computations.
Literature
We discuss the basic ideas of numerics, software and strategies in our meetings, neverthelessmany important details can’t be covered in such a short time. We recommend a basic set ofliterature in each assignment as hint for your personal studies. In terms of this assignmentwe recommend the following list of books, papers and guides:
• Finite volume methods for hyperbolic problems, R. J. LeVeque, 2002
• Riemann solvers and numerical methods for fluid dynamics, E. F. Toro, 2009
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
SWEs: Source Term
5
Prof. Dr. M. BaderDipl.-Math. A. BreuerS. Rettenberger, M. Sc.
Bachelor-Praktikum:Tsunami-Simulation April 2, 2013
In this assignment we are going to implement and test the most basic functionality of ourlab-course: The f-wave solver for the one-dimensional shallow water equations. The shallowwater equations are a system of nonlinear hyperbolic conservations laws with an optionalsource term:
h
hu
�
t
+
hu
hu2 + 1
2
gh2
�
x
= S(x, t). (1)
The quantities q = [h,hu]T are defined by h(x, t), the space-time dependent height of thewater column and hu(x, t), the space-time dependent momentum in spatial x-direction (uis the particle velocity). g the gravity constant (usually g := 9.81m/s2) and f := [hu,hu2 +1
2
gh2]T the flux function. As source term S(x, t) we will consider the e↵ect of space-dependentbathymetry only S(x) = [0,-ghB
x
]T , embedding of additional forces, such as friction or thecoriolis e↵ect is possible. Figure TODO illustrates the involved quantities.
To verify, that the most basic functionality of our program works as expected a proper(unit-) testing is required. We will do testing by a selection of standardized tests, for whicha solution is available.
Remark As units we use meters (m) and seconds (s) for all computations.
Literature
We discuss the basic ideas of numerics, software and strategies in our meetings, neverthelessmany important details can’t be covered in such a short time. We recommend a basic set ofliterature in each assignment as hint for your personal studies. In terms of this assignmentwe recommend the following list of books, papers and guides:
• Finite volume methods for hyperbolic problems, R. J. LeVeque, 2002
• Riemann solvers and numerical methods for fluid dynamics, E. F. Toro, 2009
• A wave propagation method for conservation laws and balance laws with spatially vary-
ing flux functions, D. S. Bale, 2003
• Thinking in C++: http://mindview.net/Books/TICPP/ThinkingInCPP2e.html
• git Documentation: http://git-scm.com/documentation
• Doxygen Manual : http://www.stack.nl/~dimitri/doxygen/manual
• CxxTest User Guide: http://cxxtest.com/guide.html
• SCons user Guide: http://www.scons.org/doc/production/HTML/scons-user.html
• Paraview Documentation: http://www.itk.org/Wiki/ParaView/Users_Guide/Table_Of_Contents
Monday, April 29, 13
1. ..2. F-Wave solver: Edge-local
Riemann solutions
3. ..
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
F-Waves with source term
6
Monday, April 29, 13
ghost layercomputational domain
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: Sweeps
7
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: X-Sweep
8
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: X-Sweep
9
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: X-Sweep
10
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: Y-Sweep
11
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: Y-Sweep
12
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: Sweeps
13
Q
⇤i,j
= Q
n
i,j
-�t
�x
�A
+�Q
i-1/2,j
+A
-�Q
i+1/2,j
�
x-sw
eep
y-sw
eep
= -
= -
Q
n+1
i,j
= Q
⇤i,j
-�t
�y
⇣B
+�Q
⇤i,j-1/2
+ B
-�Q
⇤i,j+1/2
⌘
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Dimensional Splitting: Stability
14
�t <
1
2· �x
�
max
x
^ �t <
1
2· �y
�
max
y
, (10)
�t = 0.4 · �x
�
max
x
, (11)
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Preparation: Hydraulic Jumps
15
Source: Hydraulics over a weir, Little River Research and Design: http://www.emriver.com
Monday, April 29, 13
Prof. Dr. M. Bader
Dipl.-Math. A. Breuer
S. Rettenberger, M. Sc.
TUM, SCCS (I5)
Bachelor-Praktikum:
Tsunami-Simulation April 29, 2013
In their di↵erential form the two dimensional shallow water equations are given by:
2
4h
hu
hv
3
5
t
+
2
4hu
hu
2 + 1
2
gh
2
huv
3
5
x
+
2
4hv
huv
hv
2 + 1
2
gh
2
3
5
y
= S(x,y, t). (1)
The quantities q = [h,hu,hv]T are defined by the space-time dependent water height
h(x,y, t), the momentum in spatial x-direction hu(x,y, t) and the momentum in y-direction
hv(x,y, t). F := [hu,hu2 + 1
2
gh
2
,huv]T is the flux function in x-direction and G :=[hv,huv,hv2+ 1
2
gh
2]T in the flux function in y-direction. g is the gravitational constant (usu-
ally g := 9.81m/s
2
). As for the one-dimensional shallow water equations, we consider only
the space-dependent e↵ect of bathymetry in the source term S(x,y) = [0,-ghB
x
,-ghB
y
]T .In this assignment we will first introduce bathymetry in our solver and simulate one di-
mensional flow over a weir in Chapter 1 and 2. The following Chapters 3 and 4 cover the
extension of our one dimensional discretization to two spatial dimensions. After this assign-
ment we have all numerics at hand and will be able to concentrate on Tsunami simulations
in the third assignment.
Literature
• Hydraulics over a weir : http://serc.carleton.edu/details/files/19076.html
• SWE Wiki : https://github.com/TUM-I5/SWE/wiki
• Documentation, GNU C Preprocessor : http://gcc.gnu.org/onlinedocs/cpp/
• Case Study: Malpasset Dam-Break Simulation using a Two-Dimensional Finite VolumeMethod, Valiani et. al., 2002
1 Bathymetry
1.1 Non-zero source term
Bathymetry is the topography of the ocean and the most important e↵ect to be considered
in the source term of the shallow water equations for Tsunami simulations. As reference we
use the sea level: Bathymetry in dry cells (land) Ci
is zero or positive: b
i
> 0, wet cells
(water) have negative values: b
i
< 0.
Introduction of bathymetry into the f-wave solver is comparably simple, because it can
be directly incorporated into the decomposition of the jump in the flux function:
�f- �x i-1/2
=2X
p=1
Z
p
, (2)
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
Preparation: Assignment 2
16
Monday, April 29, 13
Technische Universität MünchenDepartment of Informatics V
A. Breuer, S. Rettenberger, M. Bader Lab: Scientific Computing - Tsunami-Simulation
SWE
• SWE code: https://github.com/TUM-I5/SWE
17
Monday, April 29, 13