a domain decomposition approach to exponential methods for ... · exponential euler rosenbrock...
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![Page 1: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/1.jpg)
A domain decomposition approachto exponential methods for PDEs
Luca Bonaventura
MOX - Politecnico di Milano
Boulder, 8.04.2014
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 1 / 16
![Page 2: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/2.jpg)
Outline of the talk
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 3: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/3.jpg)
Outline of the talk
◮ Short review of exponential integrators
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 4: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/4.jpg)
Outline of the talk
◮ Short review of exponential integrators
◮ An accuracy and efficiency assessment of simple approaches totheir application
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 5: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/5.jpg)
Outline of the talk
◮ Short review of exponential integrators
◮ An accuracy and efficiency assessment of simple approaches totheir application
◮ Local Exponential Methods:a domain decomposition approach to exponential methods
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 6: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/6.jpg)
Outline of the talk
◮ Short review of exponential integrators
◮ An accuracy and efficiency assessment of simple approaches totheir application
◮ Local Exponential Methods:a domain decomposition approach to exponential methods
◮ Some preliminary numerical results
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 7: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/7.jpg)
Outline of the talk
◮ Short review of exponential integrators
◮ An accuracy and efficiency assessment of simple approaches totheir application
◮ Local Exponential Methods:a domain decomposition approach to exponential methods
◮ Some preliminary numerical results
◮ Conclusions and perspectives for atmospheric modelling
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 2 / 16
![Page 8: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/8.jpg)
Basic idea of exponential methods
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 3 / 16
![Page 9: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/9.jpg)
Basic idea of exponential methods
◮ Cauchy problem for nonhomogeneous linear ODE system:
du
dt= Au+ g(t) u(0) = u0
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 3 / 16
![Page 10: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/10.jpg)
Basic idea of exponential methods
◮ Cauchy problem for nonhomogeneous linear ODE system:
du
dt= Au+ g(t) u(0) = u0
◮ Representation formula for the exact solution:
u(t) = exp (At)u0 +
∫ t
0exp (A(t − s))g(s) ds
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 3 / 16
![Page 11: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/11.jpg)
Basic idea of exponential methods
◮ Cauchy problem for nonhomogeneous linear ODE system:
du
dt= Au+ g(t) u(0) = u0
◮ Representation formula for the exact solution:
u(t) = exp (At)u0 +
∫ t
0exp (A(t − s))g(s) ds
◮ Exponential methods: turn this into a numerical method witherrors indepentent of ∆t for linear problems
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 3 / 16
![Page 12: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/12.jpg)
Basic idea of exponential methods
◮ Cauchy problem for nonhomogeneous linear ODE system:
du
dt= Au+ g(t) u(0) = u0
◮ Representation formula for the exact solution:
u(t) = exp (At)u0 +
∫ t
0exp (A(t − s))g(s) ds
◮ Exponential methods: turn this into a numerical method witherrors indepentent of ∆t for linear problems
◮ Various extensions to nonlinear problems are available
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 3 / 16
![Page 13: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/13.jpg)
Exponential Euler Rosenbrock methods
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 4 / 16
![Page 14: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/14.jpg)
Exponential Euler Rosenbrock methods
◮ Linearize around initial datum at each timestep
du
dt= f(u) = f(un) + Jn(u− un) + R(u) t ∈ [tn, tn+1]
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 4 / 16
![Page 15: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/15.jpg)
Exponential Euler Rosenbrock methods
◮ Linearize around initial datum at each timestep
du
dt= f(u) = f(un) + Jn(u− un) + R(u) t ∈ [tn, tn+1]
◮ Freezing nonlinear terms yields
un+1 = un +∆tφ(
Jn∆t)
f(un) φ(z) =exp (z)− 1
z
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 4 / 16
![Page 16: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/16.jpg)
Exponential Euler Rosenbrock methods
◮ Linearize around initial datum at each timestep
du
dt= f(u) = f(un) + Jn(u− un) + R(u) t ∈ [tn, tn+1]
◮ Freezing nonlinear terms yields
un+1 = un +∆tφ(
Jn∆t)
f(un) φ(z) =exp (z)− 1
z
◮ Essentially exact for linear, constant coefficient problems,unconditionally A-stable, second order for nonlinear problems,higher order variants available (Hochbruck et al 1997)
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 4 / 16
![Page 17: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/17.jpg)
Exponential Euler Rosenbrock methods
◮ Linearize around initial datum at each timestep
du
dt= f(u) = f(un) + Jn(u− un) + R(u) t ∈ [tn, tn+1]
◮ Freezing nonlinear terms yields
un+1 = un +∆tφ(
Jn∆t)
f(un) φ(z) =exp (z)− 1
z
◮ Essentially exact for linear, constant coefficient problems,unconditionally A-stable, second order for nonlinear problems,higher order variants available (Hochbruck et al 1997)
◮ Stiff one step, one stage second order solver with oneevaluation of RHS: think of the physics...
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 4 / 16
![Page 18: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/18.jpg)
Main computational problems and solutions
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 5 / 16
![Page 19: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/19.jpg)
Main computational problems and solutions
◮ Exponential matrix cannot be stored for realistic PDE problems
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 5 / 16
![Page 20: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/20.jpg)
Main computational problems and solutions
◮ Exponential matrix cannot be stored for realistic PDE problems
◮ exp (∆tA)v can be approximated by the same Krylov spacetechniques employed in GMRES (Saad 1992)
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 5 / 16
![Page 21: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/21.jpg)
Main computational problems and solutions
◮ Exponential matrix cannot be stored for realistic PDE problems
◮ exp (∆tA)v can be approximated by the same Krylov spacetechniques employed in GMRES (Saad 1992)
◮ Krylov space dimension (and cost of time step) depend on theCourant number
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 5 / 16
![Page 22: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/22.jpg)
Main computational problems and solutions
◮ Exponential matrix cannot be stored for realistic PDE problems
◮ exp (∆tA)v can be approximated by the same Krylov spacetechniques employed in GMRES (Saad 1992)
◮ Krylov space dimension (and cost of time step) depend on theCourant number
◮ Alternative techniques imply similar costs for large scaleproblems
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 5 / 16
![Page 23: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/23.jpg)
Some numerical results
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 6 / 16
![Page 24: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/24.jpg)
Some numerical results
◮ NUMA model (courtesy of F.X.Giraldo, NPS), spatialdiscretization employing CG with fifth order polynomials
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 6 / 16
![Page 25: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/25.jpg)
Some numerical results
◮ NUMA model (courtesy of F.X.Giraldo, NPS), spatialdiscretization employing CG with fifth order polynomials
◮ Klemp-Skamarock test, Courant number approx. 23, densityfields computed by second order exponential method and BDF2at t = 400 s.
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 6 / 16
![Page 26: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/26.jpg)
Some numerical results
◮ NUMA model (courtesy of F.X.Giraldo, NPS), spatialdiscretization employing CG with fifth order polynomials
◮ Klemp-Skamarock test, Courant number approx. 23, densityfields computed by second order exponential method and BDF2at t = 400 s.
x
z
0 0.5 1 1.5 2 2.5 3
x 104
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
−1 −0.5 0 0.5 1 1.5
x 10−6
x
z
0 0.5 1 1.5 2 2.5 3
x 104
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x 10−6
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 6 / 16
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Some numerical results
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
![Page 28: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/28.jpg)
Some numerical results
◮ ICON shallow water model, low order mimetic spatialdiscretization
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
![Page 29: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/29.jpg)
Some numerical results
◮ ICON shallow water model, low order mimetic spatialdiscretization
◮ Test case 5 t = 360 h, ∆x ≈ 80 km, ∆t = 1 h, C ≈ 10
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
![Page 30: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/30.jpg)
Some numerical results
◮ ICON shallow water model, low order mimetic spatialdiscretization
◮ Test case 5 t = 360 h, ∆x ≈ 80 km, ∆t = 1 h, C ≈ 10
◮ Test case 6 at t = 240 h, ∆x ≈ 80 km, ∆t = 0.5 h C ≈ 10
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
![Page 31: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/31.jpg)
Some numerical results
◮ ICON shallow water model, low order mimetic spatialdiscretization
◮ Test case 5 t = 360 h, ∆x ≈ 80 km, ∆t = 1 h, C ≈ 10
◮ Test case 6 at t = 240 h, ∆x ≈ 80 km, ∆t = 0.5 h C ≈ 10
◮ Reference solution computed by explicit Runge Kutta methodof order 4 with ∆t = 180 s
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
![Page 32: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/32.jpg)
Some numerical results
◮ ICON shallow water model, low order mimetic spatialdiscretization
◮ Test case 5 t = 360 h, ∆x ≈ 80 km, ∆t = 1 h, C ≈ 10
◮ Test case 6 at t = 240 h, ∆x ≈ 80 km, ∆t = 0.5 h C ≈ 10
◮ Reference solution computed by explicit Runge Kutta methodof order 4 with ∆t = 180 s
h error
LP CN EX2 EX3
Test 5 1.2e-2 9.1e-3 1.2e-3 1.1e-3
Test 6 5.9e-2 1.7e-2 3.8e-4 4.0e-4
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 7 / 16
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A cost benefit analysis
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 8 / 16
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A cost benefit analysis
◮ Exponential vs highorder IMEX methods
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 8 / 16
![Page 35: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/35.jpg)
A cost benefit analysis
◮ Exponential vs highorder IMEX methods
◮ Spectral discretizationof incompressible NS -Boussinesq in sphericalgeometry (Ferran, B.,et al, JCP 2014)
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 8 / 16
![Page 36: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/36.jpg)
A cost benefit analysis
◮ Exponential vs highorder IMEX methods
◮ Spectral discretizationof incompressible NS -Boussinesq in sphericalgeometry (Ferran, B.,et al, JCP 2014)
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 8 / 16
![Page 37: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/37.jpg)
A cost benefit analysis
◮ Exponential vs highorder IMEX methods
◮ Spectral discretizationof incompressible NS -Boussinesq in sphericalgeometry (Ferran, B.,et al, JCP 2014)
10-13
10-11
10-9
10-7
10-5
10-3
10-1
10-7 10-6 10-5 10-4
ε (u)
h
a) ETDC2
ETDR2
ETDR3
ETDR4
BDF2
BDF3 BDF4
BDF5
10-13
10-11
10-9
10-7
10-5
10-3
10-1
103 104 105
ε (u)
rt
b)
ETDC2 ETDR2
ETDR3
ETDR4
BDF2 BDF3
BDF4
BDF5
BDF-VSVO
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 8 / 16
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A more local approach
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 9 / 16
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A more local approach
◮ PDEs of interest are local in space: physical and numericaldomain of dependence are finite
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 9 / 16
![Page 40: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/40.jpg)
A more local approach
◮ PDEs of interest are local in space: physical and numericaldomain of dependence are finite
◮ Local problems discretized by FD, FV, FE methods yield sparsematrices
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 9 / 16
![Page 41: A domain decomposition approach to exponential methods for ... · Exponential Euler Rosenbrock methods Linearize around initial datum at each timestep du dt = f(u) = f(un)+Jn(u −un)](https://reader030.vdocuments.site/reader030/viewer/2022041121/5f354cb838bc3a5b8e0073f8/html5/thumbnails/41.jpg)
A more local approach
◮ PDEs of interest are local in space: physical and numericaldomain of dependence are finite
◮ Local problems discretized by FD, FV, FE methods yield sparsematrices
◮ Exponential of a sparse matrix is almost sparse (Iserles 2001)
L. Bonaventura (MOX) Exponential methods Boulder, 8.04.2014 9 / 16
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A more local approach
◮ PDEs of interest are local in space: physical and numericaldomain of dependence are finite
◮ Local problems discretized by FD, FV, FE methods yield sparsematrices
◮ Exponential of a sparse matrix is almost sparse (Iserles 2001)
◮ For s−banded A = (ai ,j) with |ai ,j | ≤ ρ, let exp(A) = (ei ,j).
|ei ,j | ≤( ρs
|i − j |
)
|i−j|s[
e|i−j|s −
|i−j |−1∑
k=0
(|i − j/s|)k
k!
]
≈( ρs
|i − j |
)
|i−j|s (|i − j |/s)|i−j |
|i − j |!
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Application to PDE problems
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Application to PDE problems
◮ Advection diffusion problem: entries of matrix ∆tA scale as
u∆t
∆x+
µ∆t
∆x2
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Application to PDE problems
◮ Advection diffusion problem: entries of matrix ∆tA scale as
u∆t
∆x+
µ∆t
∆x2
◮ Example: exp(∆tA) for 1D centered finite difference advectionat Courant numbers 0.5, 5, 20
020
4060
80100
020
4060
80100
−0.4
−0.2
0
0.2
0.4
0.6
0.8
020
4060
80100
020
4060
80100
−1
−0.5
0
0.5
1
0
20
40
60
80100
020
4060
80100
−0.5
0
0.5
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Application to PDE problems
◮ Advection diffusion problem: entries of matrix ∆tA scale as
u∆t
∆x+
µ∆t
∆x2
◮ Example: exp(∆tA) for 1D centered finite difference advectionat Courant numbers 0.5, 5, 20
020
4060
80100
020
4060
80100
−0.4
−0.2
0
0.2
0.4
0.6
0.8
020
4060
80100
020
4060
80100
−1
−0.5
0
0.5
1
0
20
40
60
80100
020
4060
80100
−0.5
0
0.5
◮ There is no real need to compute a global exponential matrix:Local Exponential Methods (LEM)
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LEM: a domain decomposition approach
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LEM: a domain decomposition approach
◮ Decompose mesh in overlapping regions
M =
N⋃
i=1
Mi Mi = Di ∪ Bi
where Di non overlapping, Bi boundary buffer zones whose sizedepends on the Courant number
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LEM: a domain decomposition approach
◮ Decompose mesh in overlapping regions
M =
N⋃
i=1
Mi Mi = Di ∪ Bi
where Di non overlapping, Bi boundary buffer zones whose sizedepends on the Courant number
◮ For i = 1, . . . ,N, solve local problem restricted to Mi by a localexponential method
un+1Mi
= unMi+∆tφ
(
JnMi∆t
)
f(unMi)Mi
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LEM: a domain decomposition approach
◮ Decompose mesh in overlapping regions
M =
N⋃
i=1
Mi Mi = Di ∪ Bi
where Di non overlapping, Bi boundary buffer zones whose sizedepends on the Courant number
◮ For i = 1, . . . ,N, solve local problem restricted to Mi by a localexponential method
un+1Mi
= unMi+∆tφ
(
JnMi∆t
)
f(unMi)Mi
◮ Overwrite degrees of freedom belonging to Bi
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LEM: cons and pros
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LEM: cons and pros
◮ Overhead increases with Courant number, both forcomputation and communication...
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LEM: cons and pros
◮ Overhead increases with Courant number, both forcomputation and communication...
◮ ...but should not too bad for high order methods, anisotropicmeshes and heavy physics
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LEM: cons and pros
◮ Overhead increases with Courant number, both forcomputation and communication...
◮ ...but should not too bad for high order methods, anisotropicmeshes and heavy physics
◮ No global matrix to be computed, local problems can beparallelized trivially
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LEM: cons and pros
◮ Overhead increases with Courant number, both forcomputation and communication...
◮ ...but should not too bad for high order methods, anisotropicmeshes and heavy physics
◮ No global matrix to be computed, local problems can beparallelized trivially
◮ For small enough Di local matrices can be stored:computational gain if Jacobian is frozen every few time stepsand in the limit of large number of advected species
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A 1D numerical example
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A 1D numerical example
◮ Viscous Burgers equation with periodic boundary conditions,exact solution via Cole-Hopf transformation
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A 1D numerical example
◮ Viscous Burgers equation with periodic boundary conditions,exact solution via Cole-Hopf transformation
◮ Fourth order finite differences for advection, second order finitedifferences for diffusion, Courant number 15
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A 1D numerical example
◮ Viscous Burgers equation with periodic boundary conditions,exact solution via Cole-Hopf transformation
◮ Fourth order finite differences for advection, second order finitedifferences for diffusion, Courant number 15
◮ Second order exponential Rosenbrock method, stored localmatrices computed without Krylov spaces
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A 1D numerical example
◮ Viscous Burgers equation with periodic boundary conditions,exact solution via Cole-Hopf transformation
◮ Fourth order finite differences for advection, second order finitedifferences for diffusion, Courant number 15
◮ Second order exponential Rosenbrock method, stored localmatrices computed without Krylov spaces
0 1 2 3 4 5 6 7−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
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A 2D numerical example
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A 2D numerical example
◮ Advection-diffusion equation with rotational velocity field
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A 2D numerical example
◮ Advection-diffusion equation with rotational velocity field
◮ Monotonic finite volume method for advection, second orderfinite volume method for diffusion, Courant number 4
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A 2D numerical example
◮ Advection-diffusion equation with rotational velocity field
◮ Monotonic finite volume method for advection, second orderfinite volume method for diffusion, Courant number 4
◮ Second order exponential Rosenbrock method with localmatrices computed by Krylov space techniques
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A 2D numerical example
◮ Advection-diffusion equation with rotational velocity field
◮ Monotonic finite volume method for advection, second orderfinite volume method for diffusion, Courant number 4
◮ Second order exponential Rosenbrock method with localmatrices computed by Krylov space techniques
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50
60
70
80
90
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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A 2D nonlinear example
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A 2D nonlinear example
◮ Viscous Burgers equation
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A 2D nonlinear example
◮ Viscous Burgers equation
◮ Centered finite volume method for advection, second orderfinite volume method for diffusion, anisotropic mesh withCourant number 6 in the vertical
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A 2D nonlinear example
◮ Viscous Burgers equation
◮ Centered finite volume method for advection, second orderfinite volume method for diffusion, anisotropic mesh withCourant number 6 in the vertical
◮ Second order exponential Rosenbrock method with localmatrices computed by Krylov space techniques
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A 2D nonlinear example
◮ Viscous Burgers equation
◮ Centered finite volume method for advection, second orderfinite volume method for diffusion, anisotropic mesh withCourant number 6 in the vertical
◮ Second order exponential Rosenbrock method with localmatrices computed by Krylov space techniques
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
20 40 60 80 100 120 140 160 180 200
20
40
60
80
100
120
140
160
180
200
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
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Conclusions and perspectives
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Conclusions and perspectives
◮ Straightforward implementation of exponential methods leadsto very accurate but very costly solutions
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Conclusions and perspectives
◮ Straightforward implementation of exponential methods leadsto very accurate but very costly solutions
◮ For standard PDE problems, a local approximation ofexp (∆tA)v is feasible
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Conclusions and perspectives
◮ Straightforward implementation of exponential methods leadsto very accurate but very costly solutions
◮ For standard PDE problems, a local approximation ofexp (∆tA)v is feasible
◮ Computation of exponential matrix becomes trivially parallel
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Conclusions and perspectives
◮ Straightforward implementation of exponential methods leadsto very accurate but very costly solutions
◮ For standard PDE problems, a local approximation ofexp (∆tA)v is feasible
◮ Computation of exponential matrix becomes trivially parallel
◮ Computational overhead due to boundary buffer regions islimited in the case of anisotropic meshes and high order finiteelements
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Conclusions and perspectives
◮ Straightforward implementation of exponential methods leadsto very accurate but very costly solutions
◮ For standard PDE problems, a local approximation ofexp (∆tA)v is feasible
◮ Computation of exponential matrix becomes trivially parallel
◮ Computational overhead due to boundary buffer regions islimited in the case of anisotropic meshes and high order finiteelements
◮ Next on the to do list: use Local Exponential Methods in ahigh order FE framework and with complex forcing terms(multiple ARD with chemistry)
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