split-explicit integrators: recent developments knoth.pdf · contents 1 introduction 2...

Split-explicit integrators: Recent developments

Oswald Knoth

Leibniz Institute for Tropospheric Research, Leipzig

18.05.2017

Oswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 1 / 40

Contents

1 Introduction

2 Split-explicit methods

3 Split explicit methods, Wicker/Skamarock

4 Exponential integrator

5 Generalized split-explicit method

6 Linearized split-explicit methods

7 Numerical Examples

8 Future work


Introduction

Time scales

Time scales in atmospheric flowSound waves 340m/sGravity waves 100m/sWind speed 60m/s

Time scales with respect to spatial discretizationHorizontal-vertical grid ratioNon-aligned grid, steep orographyCut cell discretization

Use time integration methods which use different time steps or methods fordifferent scales

y ′ = f (y) + g(y)


Introduction

Time integration methods

Multirate methodsCombine explicit (and possibly implicit) methods with different timestepsSplit-explicit methodsAdditive multirate Runge-Kutta methodsStrang splitting (operator splitting)

Implicit-explicit methodsCombine explicit and implicit methods by additive partitioningPartition independent variables and/or right hand sideOne global time stepNonlinear or linear systems have to be solved

Exponential integratorAre in between the two first approaches


Split-explicit methods


First proposed by Klemp and Wilhelmson, or even earlier. Slow:Leap-frog, Fast: Forward-BackwardWicker/Skamarock (1998,2002), Slow:Runge-Kutta,Fast:Forward-Backward verticalSkamarock/Klemp (2007), Spli-Explicit in WRF: Slow:Runge-Kutta,Fast:Forward-Backward vertical, midpoint-rule verticalWicker(2009) Slow: Adams-Bashfort, Fast: forward-backwardKar(2006) Slow: Runge-Kutta, Fast: trapezoidal ruleJebens/Knoth/Weiner, Slow: Peer-method, Fast: forward-backwardWensch/Knoth (2009,2012) Generalized split-explicit methodsA lot of variants are the main time integrator in may numericalweather prediction systems



Compressible Euler equations

∂(ρv)

∂t+ ∇(ρvv)−∇τ = −∇p − ρgk− 2Ω× (ρv)

∂ρ

∂t+∇(ρv) = 0

∂(ρθ)

∂t+∇(ρθv)−∇(ρKθ∇θ) = Qθ

∂(ρµ)

∂t+∇(ρµv)−∇(ρKµ∇µ) = Qµ

p =

(Rdρθ

pκ0

)1/(1−κ)

Pressure is determined by equation of state: p = P(ρ, θ, c)

Moist potential temperature θIn the dry case θ = T

(p0p

)κwith κ ≡ Rd/cpd = const.



Splitting nonlinear equation

∂ρ

∂t= −∂ρu

∂x− ∂ρw

∂z∂ρu

∂t= −∂ρuu

∂x− ∂ρwu

∂z− ∂p

∂x∂ρw

∂t= −∂ρuw

∂x− ∂ρww

∂z− ∂p

∂z− ρg

∂ρθ

∂t= −∂ρuθ

∂x− ∂ρwθ

∂z

y = F (y , y)



Splitting nonlinear "linearized" equation

∂ρ

∂t= −∂ρu

∂x− ∂ρw

∂z∂ρu

∂t= −∂ρuu

∂x− ∂ρwu

∂z− ∂p

∂ρθ

∂ρθ

∂x∂ρw

∂t= −∂ρuw

∂x− ∂ρww

∂z− ∂p

∂ρθ

∂ρθ

∂z− ρg

∂ρθ

∂t= −∂ρuθ

∂x− ∂ρwθ

∂z

y = F (y) + A(y)y

y = F (y) + A(y)y − Ay + Ay



Splitting linearized equation

The approximate, quasi-Boussinesq linearized equations

ut = −Uux − cspx

wt = −Uwx − cspz − Nθ

θt = −Uθx − Nw

pt = −Upx − cs(ux + wz)

One dimensional acoustic advection system

ut = −Uux − cspx

pt = −Upx − csux



Split system

u = fu(u) + gp(p)

p = fp(p) + gu(u)

Slow system

u = fu(u)

p = fp(p)

Integrated wit a Runge-Kutta methodFast system

u = gp(p)

p = gu(u)

Integrated with symplectic Euler (Forward-Backward)Oswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 10 / 40

Split explicit methods, Wicker/Skamarock

Underlying Runge-Kutta

Wicker and Skamarock (MWR 2002) used a three-stage Runge-Kuttamethod as slow integrator:

un+1/3 = un +∆t

3fu(un)

pn+1/3 = pn +∆t

3fp(pn)

un+1/2 = un +∆t

2fu(un+1/3)

pn+1/2 = pn +∆t

2fp(pn+1/3)

un+1 = un + ∆tfu(un+1/2)

pn+1 = pn + ∆tfp(pn+1/2)



Split-explicit

Resulting splitting scheme:u = un, p = pn

for k = 1 : ns/3u = u + ∆τgu(p) + ∆τ fu(un)p = p + ∆τgp(u) + ∆τ fp(pn)

endun+1/3 = u, pn+1/3 = p, u = un, p = pn

for k = 1 : ns/2u = u + ∆τgu(p) + ∆τ fu(un+1/3)p = p + ∆τgp(u) + ∆τ fp(pn+1/3)

endun+1/2 = u, pn+1/2 = p, u = un, p = pn

for k = 1 : nsu = u + ∆τgu(p) + ∆τ fu(un+1/2)p = p + ∆τgp(u) + ∆τ fp(pn+1/2)

endun+1 = u, pn+1 = pOswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 12 / 40


General split-explicit methods

Assume that we can solve the fast part of

y = f (y) + g(y)

"exact"Then a split Runge–Kutta method reads:

Zi (0) = yn

Zi (τ) =1ci

i−1∑j=1

aij f (Yj) + g(Zi (τ))

Yi = Zi (cih)

yn+1 = Ys+1

For the nonlinear case

Zi (τ) =1ci

i−1∑j=1

aijF (Yj ,Zi (τ))



Stability

Test equation linear acoustic with spatial discretization and fouriermode decompositionTest equation y ′ = λy + µy , where (λ, µ) ∈ C− × C− moves over allFourier modes and CFL-numbers ,λ = +− cS ∗ Sound(exp(i ∗ l/(L + 1)) andλ = cA ∗ Adv(exp(i ∗ l/(L + 1))

Sound(z) = 1/√z −√z and Adv(z) = (2z + 3− 6/z + 1/z2)/6

Can be repaired by an over relaxed forward-backward schemeOswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 14 / 40

Exponential integrator

Phi function and linear equations

Other names: Exponential time differencing (ETD)

y = N(y) + Ly , (1)

Assume

N(y(t + h)) =m−1∑k=0

hk

k!Nk

then (1) has the solution

y(t + h) =m∑

k=1

hkφk(hL)Nk−1 + exp(hL)y(t)

with

φ0(z) = exp(z)

φk(z) =φk−1(z)− 1/(k − 1)!

z



The split explicit method

Apply the split-explicit method to (1)

Y1 = yn

Y2 = exp(1/3hL)yn + hφ1(1/3hL)N(Y1)

Y3 = exp(1/2hL)yn + hφ1(1/2hL)N(Y2)

yn+1 = exp(hL)yn + hφ1(hL)N(Y3)



The general Runge-Kutta ETD method

Yi = exp(cihL)yn + hi−1∑j=1

aij(hL)N(Yj), i = 1 . . . , s

yn+1 = exp(hL)yn + hs∑

i=1

bj(hL)N(Yj)

with

aij(hL) =K∑

k=1

αijkφk(cihL), bi (hL) =K∑

k=1

βikφk(hL)

Differential equation:

Y1 = yn

Z ′i (τ) =i−1∑j=1

K∑k=1

(τ/h)k−1

(k − 1)!

αijk

ckiN(Yj) + LZi ,Zi (0) = yn, τ ∈ [0, cih]



The method EB4 of Krogstad

012

12φ1

12

12φ1 − φ2 φ2

1 φ1 − 2φ2 0 2φ2

1 4φ3 − 3φ2 + φ1 −4φ3 + 2φ2 −4φ3 + 2φ2 4φ3 − φ2


Generalized split-explicit method

Reduction of small time step overhead

Measured in time interval length for fast problem and the actualMach-number Ma.For RK3 interval length = 1/3 + 1/2 + 1 = 11/6Numerical weather prediction models now used for weather predictionand large eddy simulationsNWP Ma = 1/6, jet stream 60 m/s, LES of the boundary layerMa = 1/20For possible large time steps , number of small time steps ≈ 100.WRF decreases large time step, such that number of small time steps< 12.Outcomes

Reduce time interval lengthLower the sound speed in the numerical schemeUse some type of implicitness




We generalize the exact integration procedure in two directions:arbitrary starting points based on preceding stages

Zni (tn + εi−1∆t) = yn +i−1∑j=1

αij(Ynj − yn)

increments in the constant term F based on preceeding stages

Zni (τ) =1δi

( i−1∑j=1

γij(Ynj − yn) +i−1∑j=1

βij f (Ynj)

)+ g(Zni (τ))

Intermediate time intervals for local integration

τ ∈ [tn + εi−1∆t, tn + εi∆t]

New set of parameters, includes standard Runge-Kutta methodsOswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 20 / 40


Order conditions

Expand numerical solution in a Taylor series.Note: Zni is a function of τ and h.3 different recursions for derivatives of Zni

four classical order conditions

bT1 =1, bT c = 1/2, bT c2 = 1/3, bTAc = 1/6

and five additional order conditionsNo 3rd order method for α = γ = 0 (classic splitting like RK3)We search for 3 stage 2rd order methodAnd for a 4 stage 3rd order methodSearch is done by solving a large nonlinear optimization problemConstraints are order conditions and stability constraintsOptimization goal: Small number of fast stepsWe found several methods



MIS4a

β21 0.38758444641450318β31, β32 -0.025318448354142823 0.38668943087310403β41, β42, β43 0.20899983523553325 -0.45856648476371231 0.43423187573425748β51, β52, β53 -0.46186171956333327 -0.46186171956333327 0.83045062122462809β54 0.27014914900250392α32, α42, α43 0.52349249922385610 1.1683374366893629 -0.75762080241712637α52, α53, α54 -0.036477233846797109 0.56936148730740477 0.47746263002599681γ32, γ42, γ43 0.13145089796226542 -0.36855857648747881 0.33159232636600550γ52, γ53, γ54 -0.065767130537473045 0.040591093109036858 0.064902111640806712

Table: Coefficients of the method MIS4a with full order 3 and 4 stages.

Integration interval length 1.43


Linearized split-explicit methods

Idea

Splitting by adding and subtracting a linear termy ′ = F (y) = Ly + (F (Y )− Ly) = Ly + f (y)

Split the linear term again in two parts L = LE + LI

A sub step in the split explicit method readsZ ′ = LEZ + LIZ + r

Apply an implicit-explicit method (IMEX) to this linear equation


Linearized split-explicit methods

IMEX-Methods

Partitioned Runge-Kutta methods for additive splittingOne method explicit, second diagonal implicit (DIRK)

c2 a21c3 a31 a32

b1 b2 b3

c1 γc2 a21 γc3 a31 a32 γ

b1 b2 b3

Zi = Z0 + ∆τi−1∑j=1

aijLEZj + ∆τi−1∑j=1

aijLIZj + ∆τ γLIZi

(I −∆τ γLI )Zi = Z0 + ∆τLE

i−1∑j=1

aijZj + ∆τLI

i−1∑j=1

aijZj

Per step two matrix vector multiplications and one linear solveOswald Knoth (TROPOS) Split-explicit integrators: Recent developments 18.05.2017 24 / 40

Numerical Examples

Straka-test with orography

Cut cell grid36000 m in the horizontal, 6400 m in the vertical, grid size 100 m

Linear part: Acoustic for the whole domain and advection near cutcellsLinear implicit part: Acoustic and advection near cut cells


Numerical Examples

Fill in structure

Fill in explicit versus implicit part for linear acoustic

Fill in implicit part for advection, scalar and vertical wind


Numerical Examples

Cold bubble


Numerical Examples

Cold bubble

Cold bubble with large time step of 1 s, 2 s, and 3 s.

dt Small time steps1 3 3 2 42 6 6 3 83 8 8 4 11


Numerical Examples

Example of Gallus/Klemp

Flow over hill of 400m of a stratified flowNonhydrostatic case ∆x = 200m and ∆x = 100mHydrostatic case ∆x = 2000m and ∆x = 100First case only cut cells implicit, ∆t = 20sNumber of small time steps, 23, 21, 11,32Second case cut cells and vertical direction for acoustic, ∆t = 100sNumber of small time steps 9, 8, 4, 12


Future work

Conclusions

Split-explicit methods are a special exponential integratorRoom for new methodsCan be used for cut cell grids with small additional overheadNumerical examples demonstrate theory


Future work

Future work

Automatic choice of implicit partEfficient implementation for parallel application

Organize computation for sub vectorsNew data structure for exchange for data

Tests for global examples


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