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

◮ 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:

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

◮ Representation formula for the exact solution:

u(t) = exp (At)u0 +

0exp (A(t − s))g(s) ds

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

u(t) = exp (At)u0 +

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

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

u(t) = exp (At)u0 +

◮ 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

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

un+1 = un +∆tφ(

Jn∆t)

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

un+1 = un +∆tφ(

Jn∆t)

◮ 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

◮ 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.

0 0.5 1 1.5 2 2.5 3

−1 −0.5 0 0.5 1 1.5

x 10−6

0 0.5 1 1.5 2 2.5 3

−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

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)

10-7 10-6 10-5 10-4

ε (u)

a) ETDC2

BDF3 BDF4

103 104 105

ε (u)

ETDC2 ETDR2

BDF2 BDF3

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)

◮ 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∑

(|i − j/s|)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

µ∆t

◮ Example: exp(∆tA) for 1D centered finite difference advectionat Courant numbers 0.5, 5, 20

−0.4

−0.2

−0.5

µ∆t

◮ Example: exp(∆tA) for 1D centered finite difference advectionat Courant numbers 0.5, 5, 20

−0.4

−0.2

−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

Mi Mi = Di ∪ Bi

where Di non overlapping, Bi boundary buffer zones whose sizedepends on the Courant number

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

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

◮ 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

0 1 2 3 4 5 6 7−0.8

−0.6

−0.4

−0.2

◮ 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

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

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)

a domain decomposition approach to exponential methods …

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