Modelado Numérico de Tsunamis
un barniz…y algunos desafíos
Patricio Catalan
Profesor Adjunto
Departamento de Obras Civiles
UTFSM
¿Por Dónde Comenzar?
Un ejercicio de modelación numérica en realidad es un ejercicio de simulación, donde el
objetivo final es lograr una reproducción lo más precisa de la realidad o fenómeno de
interés
Es un problema complejo, que cuenta al menos con tres elementos claves
3
Algoritmos y
Software
Matemática
Modelado
- Física
Ridgway Scott, PASI 2013
Objetivos:
1. Desarrollar modelos matemáticos apropiados que describan la física de los tsunami
Partamos por la física
Después de todo, el objetivo/problema es reproducir la realidad
Previo… que representa para Uds. esto:
5
@p
@x
Definición de un tsunami
Perturbación repentina de la superficie de un cuerpo de agua, de gran escala, que genera
ondas de períodos muy largos
La definición formal
Serie de ondas oceánicas de período entre 5 y 60 minutos generadas por una
perturbación a gran escala del océano.
6
2004 Indian Ocean Tsunamis
Comparisons between model results and Jason-1 measurements
(Wang and Liu, JHR 44, 2006)
(left) and TOPEX measurements (right)
Wave amplitude <0.6 m
Wavelength > 300 km
1/17/13%
15%
Ok…pero…
Esa definición no nos ayuda mucho, pero si se analizan los
datos se tiene lo siguiente
El tsunami es una onda de período muy largo…luego
su longitud es grande también:
L
L >
+100 km
Se propagan en el oceáno (principalmente), cuya
profundidad media es
h
= 4 km
Luego, la relación
h/L
es
≪ .
¿Por qué es relevante esto?
Por que en teoría de oleaje, si esa razón es pequeña, se
dice que es una onda de aguas someras
En ese caso, la velocidad de propagación de la onda es
C=√ (gh)
Resultados de campo sugieren que al menos el frente
principal se comporta así
7
C%=%Speed%of%the%leading%waves%
%%
%
C%=%212.7%m/s%and%%h%=%4.61%km%for%
Tohoku%tsunamis%
%
C%=%211.9%m/s%and%h%=%4.58%km%for%
Queen0Charlo[e%%%tsunamis%
%
C%=%190.4%%m/s%and%h%=%3.70%km%for%
Chile%tsunami%%
%
%
The%leading%wave%appears%to%
behave%like%a%linear%shallow%wave!%%
%
%
Note:%3.97km/s%%Rayleigh%wave%
speed%
%
2/ = mean depth
h C
=
g
1/17/13% 20%Algunas Ventajas de la Escala de los Tsunamis
Dos medidas de linealidad de los procesos
Profundidad/Longitud:
h/L
Muy pequeño en el oceáno abierto : lineal
Permite aproximación de aguas someras (velocidad uniforme en la
vertical)
Amplitud/Profundidad:
A/h
Relevante en aguas someras y zona de inundación : No lineal
Luego
Luego, es aparente que los tsunamis pueden ser modelados por teorías de onda larga…
¿Que significa esto?
Vamos a revisar un poco la mecánica de fluidos fundamental.
Requerimos poder representar el comportamiento cinemático y dinámico de un fluido.
Cinemático: Cómo se mueve, sin importar las fuerzas
Dinámico: Incorporando las fuerzas.
Cinemática: Conservación de la masa
10
d
dt
I
8
⇢d
8 +
I
A
⇢~v
· ˆndA = 0
D
Dt
+
· ⇥v = 0
Dinámica
Esta es fácil:
F = m*a
Dinámica: Ecuación de Navier-Stokes
Supuestos: Fluido newtoniano, flujo incompresible
12
@u
@t
+ (~u · r)~u = ~g
1
⇢
rp + ⌫r
2
~u
Aceleración
Local
Aceleración
Advectiva
Fuerza de gravedad
(por unidad de
masa)
Fuerza de superficie
debida a la presión
Fuerza de superficie
debida a la
esfuerzos de corte
de origen viscoso
El sistema así no nos es útil…
Nos aprovechamos de la condición de aguas someras
Hipótesis:
Velocidad vertical despreciable:
w=0
Velocidad horizontal prácticamente uniforme en la vertical:
u(z)=u , v(z)=0
Distribución de presión hidrostática (depende linealmente de la profundidad
solamente)
Esto permite integrar las ecuaciones entre el fondo del mar y la superficie, provistas las
condiciones de contorno siguientes
Velocidad paralela al fondo
Velocidad paralela a la superficie
Esfuerzos en la superficie despreciables
Esfuerzos de fondo representados a través de la fricción
La integración requiere del uso de la Regla de Leibniz para evaluar las cantidades en la
frontera
El sistema final
14
NON LINEAR SHALLOW WATER EQUATIONS
1
✏
@H
@t
+
r · (H~u) = O(µ
2
)
@~u
@t
+ ✏~u
· r~u + r⌘ = O(µ
2
)
CONTINUIDAD
MOMENTUM
✏ =
A
h
µ
2=
✓
h
L
◆
2H = h + ✏⌘
⌘ : Desplazamiento de la superficie libre
~u : Vector velocidad
NLSWE
EDP de tipo hiperbólico ✔
😄
Permite uso de expresiones explícitas
Puede presentar discontinuidades/shocks
Requiere de gran precisión sin oscilaciones cerca de las discontinuidades ✗
Requiere de cuidado en zonas secas/mojadas ✗
Requiere de otros términos ad-hoc
Coriolis
Rotura
15
Technology for a better society
The Shallow Water Equations
•
A Hyperbolic partial differential equation
•
Enables explicit schemes
•
Solutions form discontinuities / shocks
•
Require high accuracy in smooth parts
without oscillations near discontinuities
•
Solutions include dry areas
•
Negative water depths ruin simulations
•
Often high requirements to accuracy
•
Order of spatial/temporal discretization
•
Floating point rounding errors
•
Can be difficult to capture "lake at rest"
A standing wave or shock
27
COMMUN.MATH.SCI. ⃝c 2007 International Press
Vol. 5, No. 1, pp. 133–160
A SECOND-ORDER WELL-BALANCED POSITIVITY PRESERVING
CENTRAL-UPWIND SCHEME FOR THE SAINT-VENANT SYSTEM∗
ALEXANDER KURGANOV† AND GUERGANA PETROVA‡
Abstract. A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425]. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously.
Here, we introduce an improved second-order central-upwind scheme which, unlike its forerun-ners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.
Key words. Hyperbolic systems of conservation and balance laws, semi-discrete central-upwind schemes, Saint-Venant system of shallow water equations.
AMS subject classifications. 65M99, 35L65
1. Introduction
We are interested in developing a simple, accurate, and robust numerical method for the Saint-Venant system of shallow water equations, which was introduced more than 130 years ago in [24] and is still widely used to model flows in rivers and coastal areas. In the one-dimensional (1-D) case, the Saint-Venant system reads:
⎧ ⎨ ⎩ ht+ (hu)x= 0, (hu)t+ $ hu2+ 1 2gh 2% x= −ghB ′, (1.1)
where B(x) represents the bottom elevation, h is the fluid depth above the bottom, u is the velocity, and g is the gravitational constant.
The system (1.1) admits smooth steady-state solutions, satisfying hu = Const, u
2
2 + g(h + B) = Const,
as well as nonsmooth steady-state solutions. Both are physically relevant and thus practically important. A good numerical method for the system (1.1) should ac-curately capture both the steady states and their small perturbations (quasi-steady flows). From practical point of view, one of the most important steady-state solutions is a stationary one (lake at rest):
u = 0, h + B = Const. (1.2)
The methods that exactly preserve such solutions are called well-balanced, and we refer the reader to [2, 4, 8, 10, 12, 16, 17, 20, 21, 22, 23, 27, 28], where a variety of high-order well-balanced schemes for the Saint-Venant system can be found. Even though
∗Received: June 16, 2006; accepted (in revised version): December 9, 2006. Communicated by
Lorenzo Pareschi.
†Mathematics Department, Tulane University, New Orleans, LA 70118, (kurganov@math.tulane.
Modelo ampliado
16
Aceleración
de Coriolis
Presión no
hidrostática
Fricción de
Fondo
Difusión
Horizontal
Con esto, en principio el problema de encontrar un modelo matemático está solucionado
Algoritmos y
Software
Matemática
Modelado
¿Cómo se comportan estos modelos?
Probamos 7 modelos de uso frecuente en Chile que tienen como ecuaciones gobernantes
las presentadas anteriormente.
Para minimizar los efectos propios del tsunami, tratamos de mantener las mismas
configuraciones y dominio
Mismas condiciones iniciales
Mismo dominio de cálculo
Sin embargo, fuimos variando la resolución de la discretización
Comparación IntraModelos
18
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -6 -4 -2 0 2 4 6 Alejandro*Alcántar* Master*Student,** Departamento*de*Obras*Civiles,* Universidad*Técnica*Federico*Santa*Maria,*Valparaiso,*Chile,*Valparaiso,*Chile,* alejandro.alcantar@alumnos.usm.cl* Patricio*Catalán* Associate*Professor,** Departamento*de*Obras*Civiles,* Universidad*Técnica*Federico*Santa*Maria,*Valparaiso,*Chile,* patricio.catalan@usm.cl*
22327: Quantification of Inter-Tsunami Model
Variability for Hazard Assessment Studies
S21A-4416
Models*and*Domain
Context
Sample*InterGModel*Results
• Tsunami(modeling(is(an(essential(step(in(estimating(hazard(for(a(wide(variety(of(purposes.(For( instance,( if( the( goal( is( to( estimate( tsunami( hazard,( a( forecast( is( made( either( adopting( a( deterministic( approach( on( which( an( extreme( event( is( considered,( or( using( probabilistic( assessments.( The( latter( are( still( matter( of( active( research( and( have( not( reach( a( widespread( operational(level(yet.(
• In(this(goal,(several(models(for(tsunami(propagation(and(inundation(exist(which(are(available(for( potential(users(under(different(licensing(options.(Typically,(all(the(models(tested(are(based(on( the( Nonlinear( Shallow( Water( Equations,( although( they( might( differ( in( the( numerical( implementation(and(the(treatment(of(adChoc(terms(such(as(frictional(forces.(( • Most(of(these(models(have(been(benchmarked(against(existing(analytical,(laboratory(and(field( data(sets((Synolakis*et*al.*2008).(However,(benchmarking(is(usually(carried(out(under(optimal( conditions(in(terms(of(data(and(most(importantly,(spatial(and(temporal(resolution.( • However,(in(practice,(data(might(not(be(of(the(required(quality(in(many(areas.(This(is(especially( true(for(bathymetric(data,(which(has(been(shown(before(to(have(a(relevant(effect(on(the(results.( In(addition,(some(parameters(relevant(for(the(process(are(based(solely(on(the(decision(of(the( user.(
Objective
To( assess( the( variability( in( the( prediction( across( different( models( typically( used( in( engineering( practice.(
Methodology
1)(We(intend(to(minimize(effects(that(could(not(be(attributed(to(model(differences.(Therefore,(we( run(the(simulations(using( •A(common(bathymetry:(Composed(of(( ⁃GEBCO(data(at(30”(resolution(for(the(regional(domain.( ⁃High(resolution(bathymetry(from(Nautical(Charts(at(two(location(of(interest.( ( ( •A(common(tsunami(source,(using(a(Planar(Fault(Model(for(the(Mw(8.8(Maule(Earthquake.(The( initial(sea(level(deformation(was(estimated(by(each(model(using((their(own(internal(engines(but( using(the(same(parameters(for(the(Okada*(1985)(formulation.( 2)(We(run(several(configurations(for(each(model,(using(similar(nesting(ranging(from(coarse(model( runs(at(only(30”(resolution,(to(high(resolution(nesting(up(to(3”(at(the(point(of(interest.(The(CFL( condition(was(kept(in(the(range(0.7(for(all(cases.( •The(goal(of(this(was(to(test(the(model(variability(with(changing(grid(resolution.(3)( We( also( tested( different( model( runs( with( varying( Manning( friction( coefficient,( from( 0.010( C( 0.032( •The(goal(of(this(was(to(test(the(model(variability(with(changing(friction.( 4)(We(selected(4(target(locations(for(the(comparison.(( •Two(in(the(inner(bay(at(Valparaíso(and(Talcahuano(where(tide(gages(operated(by(the(National( Hydrographic(Service(of(the(Chilean(Navy(were(present(for(the(2010(event.( •Two(at(50(m(and(100m(deep,(outside(the(bay(to(reduce(local(bay(effects.(
C
onclusions*and*Future*Work
• Despite(sharing(the(same(fundamental(model(equations,(the(models(tested(can(show(a(large( degree(of(variability(in(the(deterministic(estimation(of(tsunami(amplitude(time(series(when(used( in((forecast(mode.( • As(expected,(this(variability(is(greatly(reduced(if(high(resolution(bathymetric(data(is(available.( However,(the(response(across(models(still(shows(a(considerable(degree(of(variability,(with(peak( amplitudes(varying(up(to(2(m((50%(of(peak(amplitude)(for(the(cases(tested.( • This(level(of(variability(across(models(have(been(observed(also(in(2D(models(based(on(the(NSWE( applied(to(Hydraulic(Modeling((e.g.(Neelz*and*Pender,*2010)(• Future( work( considers( expanding( the( analysis( to( deeper( water,( to( minimize( interpolation(( artifacts(and((resonant(effects.( • Also,(comparison(with(actual(tsunami(time(series((available(from(the(Mw8.1(Iquique(Earthquake( will(be(carried(out.(
Acknowledgements
This(work(has(been(funded(by(the(Chilean(National(Science(and(technology(Committee(CONICYT( through(FONDEF(Grant((D11I1119(and(CONICYT/FONDAP/15110017(program(CIGIDENWe( tested( 7( models( which( are( commonly( used( in( Chile( either( by( scientists( and( engineers.(These(are( •TUNAMI( •COMCOT(( •NEOWAVE(( •GeoClaw(( •TsunAWI(( •MIKE21((( •Delft3D
Results
Sample*IntraGmodel*Results
These(figures(show(all(the(time(series(at(the(tide(gage(for(a(given(model(under(changing(bathymetric(size(and(fixed(Manning(roughness( coefficient((thin(lines).(The(dark(lines(are(the(average(time(series(and(the(dotted(line(is(the(standard(deviation(among(runs.( IntraCmodel(results(could(be(grouped(in(three(typical(model(response(to(changes(in(grid(resolution:( Case B • (Case*A)(High(sensitivity(in(free(surface(displacement(but(negligible(dependency( on( phase( (arrival( time).( Variability,( estimated( by( the( standard(deviation,(was(comparable(to(the(average(signal.(
• 2(models((fell(into(this(group.
• (Case* B)( Large( sensitivity( in( free( surface( displacement( and( a(
notorious( dependency( on( phase( (arrival( time).( Variability( was( comparable(to(the(average(signal.( • 2(models(fell(into(this(group.( tide gages Longitude, W La titude , S
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -6 -4 -2 0 2 4 6 Case A
• (Case* C)( Reduced( sensitivity( in( free( surface( displacement( but( ( it(
increases( significantly( at( crest( and( troughs.( Variability( was( greatly( reduced( when( compared( to( other( cases,( showing( a( remarkable( consistency(in(the(phase.(
• 3models(fell(into(this(group.(
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -6 -4 -2 0 2 4 6 Case C These(figures(show(the(time(series(of(all(models(for(a(fixed(Manning(roughness(factor(and(comparable(grid( settings.(
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -10 -8 -6 -4 -2 0 2 4 6 8 10
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -10 -8 -6 -4 -2 0 2 4 6 8 10
• (Coarse* Grid)( A( high( degree( of( variability( is(
observed( both( in( amplitude( and( arrival( times.( Peak( amplitudes( vary( between( 1C6,( with( large( variations(in(temporal(response.((
• Overall(structure(of(the(time(series(is(controlled(by( resonant(conditions(at(bay(level.
• (Finer* Grid)( Most( models( tend( to( yield( a( similar(
temporal( response,( with( comparable( peak(( amplitudes( and( phases,( although( the( location( of( the(maxima(can(vary(significantly(in(time.((
• A( couple( of( models( deviate( notoriously,( which( is( under(further(investigation.
References
Néelz,(S.(&(Pender,(G.((2010),('Benchmarking(of(2D(Hydraulic(Modelling(Packages'(SCHO0510BSNOCECP),(Technical(report,(Environment( Agency(Science(Report(,SC080035/R2,(Rio(House,(Waterside(Drive,(Aztec(West,(Almondsbury,(Bristol,(BS32(4UD.( Synolakis,(C.;(Bernard,(E.;(Titov,(V.;(Kвnoglu,(U.(&(González,(F.((2008),('Validation(and(verification(of(tsunami(numerical(models',(Pure(and( Applied(Geophysics(165(11),(2197CC2228.(Comparación InterModelos
19
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -6 -4 -2 0 2 4 6
Alejandro*Alcántar*
Master*Student,**
Departamento*de*Obras*Civiles,*
Universidad*Técnica*Federico*Santa*Maria,*Valparaiso,*Chile,*Valparaiso,*Chile,*
alejandro.alcantar@alumnos.usm.cl*
Patricio*Catalán*
Associate*Professor,**
Departamento*de*Obras*Civiles,*
Universidad*Técnica*Federico*Santa*Maria,*Valparaiso,*Chile,*
patricio.catalan@usm.cl*
22327: Quantification of Inter-Tsunami Model
Variability for Hazard Assessment Studies
S21A-4416
Models*and*Domain
Context
Sample*InterGModel*Results
•
Tsunami(modeling(is(an(essential(step(in(estimating(hazard(for(a(wide(variety(of(purposes.(For(
instance,( if( the( goal( is( to( estimate( tsunami( hazard,( a( forecast( is( made( either( adopting( a(
deterministic( approach( on( which( an( extreme( event( is( considered,( or( using( probabilistic(
assessments.( The( latter( are( still( matter( of( active( research( and( have( not( reach( a( widespread(
operational(level(yet.(
•
In(this(goal,(several(models(for(tsunami(propagation(and(inundation(exist(which(are(available(for(
potential(users(under(different(licensing(options.(Typically,(all(the(models(tested(are(based(on(
the( Nonlinear( Shallow( Water( Equations,( although( they( might( differ( in( the( numerical(
implementation(and(the(treatment(of(adChoc(terms(such(as(frictional(forces.((
•
Most(of(these(models(have(been(benchmarked(against(existing(analytical,(laboratory(and(field(
data(sets((Synolakis*et*al.*2008).(However,(benchmarking(is(usually(carried(out(under(optimal(
conditions(in(terms(of(data(and(most(importantly,(spatial(and(temporal(resolution.(
•
However,(in(practice,(data(might(not(be(of(the(required(quality(in(many(areas.(This(is(especially(
true(for(bathymetric(data,(which(has(been(shown(before(to(have(a(relevant(effect(on(the(results.(
In(addition,(some(parameters(relevant(for(the(process(are(based(solely(on(the(decision(of(the(
user.(
Objective
To( assess( the( variability( in( the( prediction( across( different( models( typically( used( in( engineering(
practice.(
Methodology
1)(We(intend(to(minimize(effects(that(could(not(be(attributed(to(model(differences.(Therefore,(we(
run(the(simulations(using(
•A(common(bathymetry:(Composed(of((
⁃GEBCO(data(at(30”(resolution(for(the(regional(domain.(
⁃High(resolution(bathymetry(from(Nautical(Charts(at(two(location(of(interest.(
(
(
•A(common(tsunami(source,(using(a(Planar(Fault(Model(for(the(Mw(8.8(Maule(Earthquake.(The(
initial(sea(level(deformation(was(estimated(by(each(model(using((their(own(internal(engines(but(
using(the(same(parameters(for(the(Okada*(1985)(formulation.(
2)(We(run(several(configurations(for(each(model,(using(similar(nesting(ranging(from(coarse(model(
runs(at(only(30”(resolution,(to(high(resolution(nesting(up(to(3”(at(the(point(of(interest.(The(CFL(
condition(was(kept(in(the(range(0.7(for(all(cases.(
•The(goal(of(this(was(to(test(the(model(variability(with(changing(grid(resolution.(
3)( We( also( tested( different( model( runs( with( varying( Manning( friction( coefficient,( from( 0.010( C(
0.032(
•The(goal(of(this(was(to(test(the(model(variability(with(changing(friction.(
4)(We(selected(4(target(locations(for(the(comparison.((
•Two(in(the(inner(bay(at(Valparaíso(and(Talcahuano(where(tide(gages(operated(by(the(National(
Hydrographic(Service(of(the(Chilean(Navy(were(present(for(the(2010(event.(
C
onclusions*and*Future*Work
•
Despite(sharing(the(same(fundamental(model(equations,(the(models(tested(can(show(a(large(
degree(of(variability(in(the(deterministic(estimation(of(tsunami(amplitude(time(series(when(used(
in((forecast(mode.(
•
As(expected,(this(variability(is(greatly(reduced(if(high(resolution(bathymetric(data(is(available.(
However,(the(response(across(models(still(shows(a(considerable(degree(of(variability,(with(peak(
amplitudes(varying(up(to(2(m((50%(of(peak(amplitude)(for(the(cases(tested.(
•
This(level(of(variability(across(models(have(been(observed(also(in(2D(models(based(on(the(NSWE(
applied(to(Hydraulic(Modeling((e.g.(Neelz*and*Pender,*2010)(
•
Future( work( considers( expanding( the( analysis( to( deeper( water,( to( minimize( interpolation((
artifacts(and((resonant(effects.(
•
Also,(comparison(with(actual(tsunami(time(series((available(from(the(Mw8.1(Iquique(Earthquake(
will(be(carried(out.(
Acknowledgements
This(work(has(been(funded(by(the(Chilean(National(Science(and(technology(Committee(CONICYT(
through(FONDEF(Grant((D11I1119(and(CONICYT/FONDAP/15110017(program(CIGIDEN
We( tested( 7( models( which( are( commonly(
used( in( Chile( either( by( scientists( and(
engineers.(These(are(
•TUNAMI(
•COMCOT((
•NEOWAVE((
•GeoClaw((
•TsunAWI((
•MIKE21(((
•Delft3D
Results
Sample*IntraGmodel*Results
These(figures(show(all(the(time(series(at(the(tide(gage(for(a(given(model(under(changing(bathymetric(size(and(fixed(Manning(roughness(
coefficient((thin(lines).(The(dark(lines(are(the(average(time(series(and(the(dotted(line(is(the(standard(deviation(among(runs.(
IntraCmodel(results(could(be(grouped(in(three(typical(model(response(to(changes(in(grid(resolution:(
Case B
•
(Case*A)(High(sensitivity(in(free(surface(displacement(but(negligible(
dependency( on( phase( (arrival( time).( Variability,( estimated( by( the(
standard(deviation,(was(comparable(to(the(average(signal.(
•
2(models((fell(into(this(group.
•
(Case* B)( Large( sensitivity( in( free( surface( displacement( and( a(
notorious( dependency( on( phase( (arrival( time).( Variability( was(
comparable(to(the(average(signal.(
•
2(models(fell(into(this(group.(
tide
gages
Longitude, W
La
titude
, S
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -6 -4 -2 0 2 4 6
Case A
•
(Case* C)( Reduced( sensitivity( in( free( surface( displacement( but( ( it(
increases( significantly( at( crest( and( troughs.( Variability( was( greatly(
reduced( when( compared( to( other( cases,( showing( a( remarkable(
consistency(in(the(phase.(
•
3models(fell(into(this(group.(
η [m] -6 -4 -2 0 2 4 6Case C
These(figures(show(the(time(series(of(all(models(for(a(fixed(Manning(roughness(factor(and(comparable(grid(
settings.(
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -10 -8 -6 -4 -2 0 2 4 6 8 10
Elapsed Time [Min]
0 50 100 150 200 250 300 η [m] -10 -8 -6 -4 -2 0 2 4 6 8 10
•
(Coarse* Grid)( A( high( degree( of( variability( is(
observed( both( in( amplitude( and( arrival( times.(
Peak( amplitudes( vary( between( 1C6,( with( large(
variations(in(temporal(response.((
•
Overall(structure(of(the(time(series(is(controlled(by(
resonant(conditions(at(bay(level.
•
(Finer* Grid)( Most( models( tend( to( yield( a( similar(
temporal( response,( with( comparable( peak((
amplitudes( and( phases,( although( the( location( of(
the(maxima(can(vary(significantly(in(time.((
•
A( couple( of( models( deviate( notoriously,( which( is(
under(further(investigation.
References
Néelz,(S.(&(Pender,(G.((2010),('Benchmarking(of(2D(Hydraulic(Modelling(Packages'(SCHO0510BSNOCECP),(Technical(report,(Environment(
Agency(Science(Report(,SC080035/R2,(Rio(House,(Waterside(Drive,(Aztec(West,(Almondsbury,(Bristol,(BS32(4UD.(
¿Cómo se Explica Esto?
20
Algoritmos y
Software
Matemática
Modelado
- Física
Implementación Numérica
El desafío es pasar de un esquema continuo (como las derivadas parciales), a un esquema
discreto como lo es una malla de cálculo numérica
Sin perder precisión
Con bajo costo
Sin introducir errores
La ecuación es hiperbólica y resuelta como un problema de condición inicial o de contorno.
El método típico es de marcha, en el espacio y el tiempo.
La forma de resolver la ecuación parcial en diferencias finitas puede tener efectos en la
respuesta del problema
Difusión Numérica
Convección Numérica
Dispersión Numérica
21
Universidad Técnica Federico Santa María FONDEF D11I1119
Ecuación de Aguas Someras: Métodos de resolución
Se resumirá en la siguiente tabla los métodos empleados para ambos la CPU y GPU.
CPU
GPU
Comcot
Tunami
Neowave
Tusunawi
GeoClaw
SWE
Formulation Conservative Conservative
Conservative Conservative
Non-
Non-
Conservative Conservative
Method
Difference
Finite
Difference
Finite
Difference
Finite
Elements
Finite
Difference
Finite
Difference
Finite
Scheme
Explicit
Explicit
Implicit
Explicit
Explicit
Explicit
Tab. 2: Análisis de los métodos empleados en cada Software.
Discusión:
Es importante destacar que el método explícitos de diferencias finitas son preferibles porque sólamente evaluaciones
en cada paso de tiempo son requeridas. También es fácil de ver que los esquemas de diferencia finitas son mas populares
para la evolución en el dominio del tiempo y el espacio. La razón principal se puede atribuir al hecho de que los sistemas
de diferencia finitas son altamente paralelizables y pueden ser utilizada con openMP para la CPU y openCL or CUDA
para la GPU, (e.g. Brodtkorb et al. (2010, 2012)). A demás, como la disminución en el tiempo requerido para completar
las simulaciónes es uno de nuestro objetivo principales, un esquema de diferencia finitas explícito debería ser el enfoque
que debe adoptarse.
Ecuación de Aguas Someras: Esquemas
En la siguiente tabla se resumen los esquemas de solución para la CPU y GPU empleadas tanto para la continuidad y la
ecuación de momentum. El concepto de esquema guarda relación con la forma en que se realiza la discretización de las
variables y operadores para pasar de una variable continua, como por ejemplo, el nivel del mar en el caso de un tsunami, a
una variable discreta, como lo sería el nivel del mar calculado ahora en los puntos de la malla de cálculo. La definición de
esto es fundamental, pues distintos esquemas de cálculo pueden favorecer distintos aspectos, como por ejemplo, la
estabili-dad numérica y/o la propagación de errores de truncación. Adicionalmente, en la tabla también se incluye algunas formas
adoptadas paralela aumentar la la precisión conforme se acerca la onda a las zonas de interés como bahías y ciudades.:
CPU
GPU
Comcot
Tunami
Neowave
TsunAWI
GeoClaw
SWE
Continuity
Leap-frog
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Momentum
First-order
Upwind
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Accuracy
Nested-
Grids
Nested-
Grids
Refinement
Mesh
Adaptative Mesh
Refinement
Time-step
Variable
Tab. 3: Análisis de los esquemas empleados por cada Software.
Discusión:
Diferencia Central
Ventajas:
Es más sencillo de programar y requieren menos tiempo en la computadora por paso de
tiempo. ⋄
El esquema es más preciso que un Central–Upwind de primer orden.
Un parámetro de disipación es necesario para acercarse a un estado de equilibrio.
Permite el uso de múltiples grillas (grillas anidadas) para regiones extensas.
Desventajas:
Es un método altamente inestable.
Es algo más disipativo de manera numérica, lo que significa que la onda decae debido a
la precisión del método y no sólo debido a procesos físicos.
Tiene que ser modificado para obtener las propiedades de segundo orden, para mejorar
la precisión.
Depende en gran medida del
∆t
y la discretización espacial
∆x
.
Esquemas numéricos
22
Universidad Técnica Federico Santa María FONDEF D11I1119
Ecuación de Aguas Someras: Métodos de resolución
Se resumirá en la siguiente tabla los métodos empleados para ambos la CPU y GPU.
CPU
GPU
Comcot
Tunami
Neowave
Tusunawi
GeoClaw
SWE
Formulation Conservative Conservative
Conservative Conservative
Non-
Non-
Conservative Conservative
Method
Difference
Finite
Difference
Finite
Difference
Finite
Elements
Finite
Difference
Finite
Difference
Finite
Scheme
Explicit
Explicit
Implicit
Explicit
Explicit
Explicit
Tab. 2: Análisis de los métodos empleados en cada Software.
Discusión:
Es importante destacar que el método explícitos de diferencias finitas son preferibles porque sólamente evaluaciones
en cada paso de tiempo son requeridas. También es fácil de ver que los esquemas de diferencia finitas son mas populares
para la evolución en el dominio del tiempo y el espacio. La razón principal se puede atribuir al hecho de que los sistemas
de diferencia finitas son altamente paralelizables y pueden ser utilizada con openMP para la CPU y openCL or CUDA
para la GPU, (e.g. Brodtkorb et al. (2010, 2012)). A demás, como la disminución en el tiempo requerido para completar
las simulaciónes es uno de nuestro objetivo principales, un esquema de diferencia finitas explícito debería ser el enfoque
que debe adoptarse.
Ecuación de Aguas Someras: Esquemas
En la siguiente tabla se resumen los esquemas de solución para la CPU y GPU empleadas tanto para la continuidad y la
ecuación de momentum. El concepto de esquema guarda relación con la forma en que se realiza la discretización de las
variables y operadores para pasar de una variable continua, como por ejemplo, el nivel del mar en el caso de un tsunami, a
una variable discreta, como lo sería el nivel del mar calculado ahora en los puntos de la malla de cálculo. La definición de
esto es fundamental, pues distintos esquemas de cálculo pueden favorecer distintos aspectos, como por ejemplo, la
estabili-dad numérica y/o la propagación de errores de truncación. Adicionalmente, en la tabla también se incluye algunas formas
adoptadas paralela aumentar la la precisión conforme se acerca la onda a las zonas de interés como bahías y ciudades.:
CPU
GPU
Comcot
Tunami
Neowave
TsunAWI
GeoClaw
SWE
Continuity
Leap-frog
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Momentum
First-order
Upwind
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Accuracy
Nested-
Grids
Nested-
Grids
Refinement
Mesh
Adaptative Mesh
Refinement
Time-step
Variable
Tab. 3: Análisis de los esquemas empleados por cada Software.
Discusión:
MSc. Danilo S. Kusanovic, Patricio Catalán
Tsumami Modelling
7/11
65
Fig. 19 Two-dimensional structured mesh for finite difference approximations.
∂
U
∂
x
!
!
!
!
i, j≈
U
i+1, j−
U
i−1, j2
∆
x
≡
δ
2xU
i, j.
∂
U
∂
y
!
!
!
!
i, j≈
U
i, j+1−
U
i, j−12
∆
y
≡
δ
2yU
i, j.
In general, finite difference approximation will involve a stencil of points surrounding U
i, j.
65
Fig. 19 Two-dimensional structured mesh for finite difference approximations.
∂U ∂ x ! ! ! ! i, j ≈ Ui+1, j−Ui−1, j 2∆ x ≡δ2xUi, j. ∂U ∂ y ! ! ! ! i, j ≈ Ui, j+1−Ui, j−1 2∆ y ≡δ2yUi, j.
Esquemas numéricos
23
Universidad Técnica Federico Santa María FONDEF D11I1119
Ecuación de Aguas Someras: Métodos de resolución
Se resumirá en la siguiente tabla los métodos empleados para ambos la CPU y GPU.
CPU
GPU
Comcot
Tunami
Neowave
Tusunawi
GeoClaw
SWE
Formulation Conservative Conservative
Conservative Conservative
Non-
Non-
Conservative Conservative
Method
Difference
Finite
Difference
Finite
Difference
Finite
Elements
Finite
Difference
Finite
Difference
Finite
Scheme
Explicit
Explicit
Implicit
Explicit
Explicit
Explicit
Tab. 2: Análisis de los métodos empleados en cada Software.
Discusión:
Es importante destacar que el método explícitos de diferencias finitas son preferibles porque sólamente evaluaciones
en cada paso de tiempo son requeridas. También es fácil de ver que los esquemas de diferencia finitas son mas populares
para la evolución en el dominio del tiempo y el espacio. La razón principal se puede atribuir al hecho de que los sistemas
de diferencia finitas son altamente paralelizables y pueden ser utilizada con openMP para la CPU y openCL or CUDA
para la GPU, (e.g. Brodtkorb et al. (2010, 2012)). A demás, como la disminución en el tiempo requerido para completar
las simulaciónes es uno de nuestro objetivo principales, un esquema de diferencia finitas explícito debería ser el enfoque
que debe adoptarse.
Ecuación de Aguas Someras: Esquemas
En la siguiente tabla se resumen los esquemas de solución para la CPU y GPU empleadas tanto para la continuidad y la
ecuación de momentum. El concepto de esquema guarda relación con la forma en que se realiza la discretización de las
variables y operadores para pasar de una variable continua, como por ejemplo, el nivel del mar en el caso de un tsunami, a
una variable discreta, como lo sería el nivel del mar calculado ahora en los puntos de la malla de cálculo. La definición de
esto es fundamental, pues distintos esquemas de cálculo pueden favorecer distintos aspectos, como por ejemplo, la
estabili-dad numérica y/o la propagación de errores de truncación. Adicionalmente, en la tabla también se incluye algunas formas
adoptadas paralela aumentar la la precisión conforme se acerca la onda a las zonas de interés como bahías y ciudades.:
CPU
GPU
Comcot
Tunami
Neowave
TsunAWI
GeoClaw
SWE
Continuity
Leap-frog
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Momentum
First-order
Upwind
Difference
Central-
First-order
Upwind
Leap-frog
First-order
Upwind
Central-
Upwind
Accuracy
Nested-
Grids
Nested-
Grids
Refinement
Mesh
Adaptative Mesh
Refinement
Time-step
Variable
Tab. 3: Análisis de los esquemas empleados por cada Software.
Discusión:
MSc. Danilo S. Kusanovic, Patricio Catalán
Tsumami Modelling
7/11
65
Fig. 19 Two-dimensional structured mesh for finite difference approximations.
∂U ∂ x ! ! ! ! i, j ≈ Ui+1, j−Ui−1, j 2∆ x ≡δ2xUi, j. ∂U ∂ y ! ! ! ! i, j ≈ Ui, j+1−Ui, j−1 2∆ y ≡δ2yUi, j.
In general, finite difference approximation will involve a stencil of points surrounding Ui, j.
62
In the following we derive a finite difference approximation of
∂U
∂ x
at node x
i. (Note: this is essentially the same
procedure as was used to derive the midpoint method in Chapter 3). For a differentiable function U, the derivative at
the point x
iis given by:
∂U
∂ x
!
!
!
!
xi= lim
∆x→0U(x
i+ ∆ x) −U(x
i−
∆ x)
2∆ x
(95)
The finite difference approximation is obtained by eliminating the limiting process:
U
xi≈
U(x
i+ ∆ x) −U(x
i−
∆ x)
2∆ x
=
U
i+1−
U
i−12∆ x
≡
δ
2xU
i.
(96)
The finite difference operator δ
2xis called a central difference operator. Finite difference approximations can also be
one-sided. For example, a backward difference approximation is,
U
xi≈
1
∆ x
(U
i−
U
i−1) ≡ δ
−
x
U
i,
(97)
and a forward difference approximation is,
U
xi≈
1
∆ x
(U
i+1−
U
i) ≡ δ
+
x
