A domain decomposition approach to exponential methods …

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

Outline of the talk


Outline of the talk

◮ Short review of exponential integrators


Outline of the talk


◮ An accuracy and efficiency assessment of simple approaches totheir application


Outline of the talk



◮ Local Exponential Methods:a domain decomposition approach to exponential methods


Outline of the talk




◮ Some preliminary numerical results


Outline of the talk




◮ Some preliminary numerical results

◮ Conclusions and perspectives for atmospheric modelling


Basic idea of exponential methods



◮ Cauchy problem for nonhomogeneous linear ODE system:

du

dt= Au+ g(t) u(0) = u0




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




du

dt= Au+ g(t) u(0) = u0


u(t) = exp (At)u0 +

∫ t


◮ Exponential methods: turn this into a numerical method witherrors indepentent of ∆t for linear problems




du

dt= Au+ g(t) u(0) = u0


u(t) = exp (At)u0 +

∫ t


◮ Exponential methods: turn this into a numerical method witherrors indepentent of ∆t for linear problems

◮ Various extensions to nonlinear problems are available


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]




du


◮ Freezing nonlinear terms yields

un+1 = un +∆tφ(

Jn∆t)

f(un) φ(z) =exp (z)− 1

z




du



un+1 = un +∆tφ(

Jn∆t)


z

◮ Essentially exact for linear, constant coefficient problems,unconditionally A-stable, second order for nonlinear problems,higher order variants available (Hochbruck et al 1997)




du



un+1 = un +∆tφ(

Jn∆t)


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...


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





◮ Krylov space dimension (and cost of time step) depend on theCourant number

◮ Alternative techniques imply similar costs for large scaleproblems


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.




◮ 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





◮ 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






◮ 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


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


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)





◮ 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 |!


Application to PDE problems



◮ Advection diffusion problem: entries of matrix ∆tA scale as

u∆t

∆x+

µ∆t

∆x2




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




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)


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




M =

N⋃

i=1

Mi Mi = Di ∪ Bi


◮ For i = 1, . . . ,N, solve local problem restricted to Mi by a localexponential method

un+1Mi

= unMi+∆tφ

(

JnMi∆t

)

f(unMi)Mi




M =

N⋃

i=1

Mi Mi = Di ∪ Bi


◮ 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


LEM: cons and pros


LEM: cons and pros

◮ Overhead increases with Courant number, both forcomputation and communication...


LEM: cons and pros


◮ ...but should not too bad for high order methods, anisotropicmeshes and heavy physics


LEM: cons and pros



◮ No global matrix to be computed, local problems can beparallelized trivially


LEM: cons and pros



◮ 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


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





◮ 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





◮ 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


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











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


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






◮ 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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A domain decomposition approach to exponential methods …