On perspectives for the numericalSolution of the WAE: Experiments and

Decomposition Methods

Aron Roland and Jürgen GeiserTechnische Universität Darmstadt

Humboldt Universität Berlin

WISE 2008, Helsinki, Finland

Overview of the talk

1. Available numerical schemes for the solution of the WAE

2. Operator Splitting (OSM) and Iterative Direct (IDM) Methods

3. Splitting error demonstrated on simple cases

4. Some comparison between IDM (SWAN) and OSM (WWM)

5. Perspectives for the solution of the WAE on unstructured meshes using OSM

• Splitting errors of classical methods (Splitting Errors)• Iterative Operator Splitting methods (Skip Splitting Errors)• Benefits of iterative methods (Efficiency, Simplicity)• Adaptive time-decomposition using the eigenvalues

The Wave Action Equation (WAE)

Total Source TermAdvection in geographical spaceChange in Time Advection in spectral space

totN N N N St

X

X

A

d d

dt d

X g

XX = c c U

k

1 Adc

k d m s

Uk

AA g

dc d c

d t s

X

UU k

4 3 ...tot in nl ds nl br bf bgS S S S S S S S

• Implicit Iterative Direct Methods (SWAN)

– Discretization of the whole equation as suggested by Patankar (1980) see Booij et al. (1999) for the WAE

Numerical schemesIterative Direct Methods (IDM)

Total Source TermAdvection in geographical spaceChange in Time Advection in spectral space

totN N N N St

X

X

• Operator Splitting Methods (OSM) e.g.: WAM, MIKE21, TOMAWAC, CREST …

– 1st Step – Advection

– 2nd Step – Source Terms

*

*000 ; on 0,x y t

Nc N c N c N c N N N t

t x y

**

** *, 0; on 0,totN t t t

NS N N t

t

Numerical schemesOperator Splitting Methods

*

( ) *000 ; on 0,t

DNN N t

Dt

k

• Operator Splitting Methods (OSM) e.g. WWIII or WWM

– 1st Step – Spectral part

– 2nd Step – Geographical space

– 3rd Step – Integration of the source terms

*

*000; on 0,t

Nc N N N t

t

***

** ** *** **00 ; on 0,x y t t t

Nc N c N N N t

t x y

****

**** ***,** 0; on 0,totN t t t

NS N N t

t

Numerical schemesOperator Splitting Methods II

**

* ** *00; on 0,t t t

Nc N N N t

t

• Operator Splitting Methods and Iterative Sources (OSM-IS); WWM

– 1st Step – Spectral part

– 2nd Step – Geographical space and sources

*

*000; on 0,t

Nc N N N t

t

***

** ** *** **,** 0 ; on 0,x y totN t t t

Nc N c N S N N t

t x y

Numerical schemesOperator Splitting Methods III

**

* ** *00; on 0,t t t

Nc N N N t

t

Two important mathematical theorems

• The Godunov theorem– Linear schemes (linear combination of the unknown variables) cannot be

monotone and higher order!

• The Lax theorem– The numerical scheme is convergent if it can be proven that it is stable and

consistent.

• Consequences:– Implicit schemes must be nonlinear to achieve higher order accuracy while

retaining monotonicity. This leads directly to nonlinear equation systems that have to be solved

– Explicit methods can be easily defined to be nonlinear and achieve in this way higher order accuracy though retaining monotonicity.

– The convergence of OSM methods is much easier to prove as for the direct methods since the chosen scheme for the solution of the sub-problems are in most cases shown to be consistent and stable by the authors itself.

Numerical schemes in the WWM II

• Numerical methods for the sub-problems– Geographical space

• Galerkin schemes (non-monotone, conservative)– Crank-Nicolson Taylor Galerkin (implicit, 2nd order in space and time)– Euler Taylor Galerkin (implicit, 2nd order in space)

• Residual distribution schemes (monotone, conservative)– CRD-N1 schemes (implicit)– CRD-N3 schemes (implicit, 2nd order in time)– CRD-N scheme (explicit)– CRD-PSI scheme (explicit, better then 1st order)– CRD-FCT scheme (explicit, 2nd order in space and time)

– Source term integration • Semi-implicit (WAM)• Dynamical (WWIII)• Iteratively within the implicit advection schemes in geographical

space

– Spectral space• Ultimate Quickest (explicit, 3rd order in space and time)

Splitting Error between advection and strong local sources

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00Length [m]

Hs

[m]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Dep

th [

m]

Hs (dt=1.0) NEXPDYN

Hs (dt=0.005)

Hs (dt=1.0) NIMPDYN

Depth

Left: Significant wave height along a cross section for the unsplitted solution (blue) with Δt = 0.005 compared to the splitted solution using the explicit CRD-N scheme (red) and the implicit CRD-N1 scheme (green)

CFLX=14

For Δt=1.0s

0.00

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Global time step [s]

L2

Err

or

No

rm [

m]

CRD-N3IMP_SI

CRD-N3IMP_UNSPLIT

CRD-N3IMP_DYN

0.00

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Global time step [s]

L2

Err

or

No

rm [

m]

CRD-N1IMP_SI

CRD-N1IMP_UNSPLIT

CRD-N1IMP_DYN

CNTG_UNSPLIT

0.00

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Global time step [s]

L2

Err

or

No

rm [

m]

CRD-N1IMP_UNSPLIT

CRD-N1EXP_SI

CRD-N1EXP_DYN

Splitting Error between advection and strong local sources

2 21max

1 1 m

ii

en

L 2 2

1max

1 1 m

ii

en

L

2 21max

1 1 m

ii

en

L

0.00

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Global time step [s]

L2

Err

or

No

rm [

m]

CRD-N1IMP_UNSPLIT

CRD-FCTEXP_SI

CRD-FCTEXP_DYN 2 21max

1 1 m

ii

en

L

Splitting Error Advection in Geographical and spectral space

Bathymetry (left) and Computational mesh (right).

Splitting Error Advection in Geographical and spectral space

Wave height (left) and Average wave direction (right).

Splitting Error Advection in Geographical and spectral space

CFLX=11

CFLθ=3.2

For Δt=1.0s

Splitting Error Advection in Geographical and spectral space

CFLX=11

CFLθ=3.2

For Δt=1.0s

Splitting Error Advection in Geographical and spectral space

CFLX=11

CFLθ=3.2

For Δt=1.0s

Splitting Error between advection in geographical space and spectral space

0.00

0.02

0.04

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Global time step [s]

L2 E

rror

Nor

m [m

]

CNTG-CN THETA = 0.5

CNTG-CN THETA = 1.0

0.00

0.02

0.04

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Global time step [s]

L2 E

rror

Nor

m [m

]

CNTGCRD-N1EXPCRD-FCTCRD-N1IMPCRD-N2IMPCRD-N3IMP

L2 error norm of the CNTG and ETG scheme in combination with the CN and the EI scheme for directional space respectively (left). L2 error norm for the CNTG, the CRD-N1EXP, the CRD-FCT the CRD-N1IMP, the CRD-N2IMP and the CRD-N3IMP scheme using the UQ approach for the integration in directional space (right).

Continental Shelf refraction (SWAN)

0

5

10

15

20

25

30

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000Length [m]

Dire

ctio

nal s

prea

ding

[°]

-200

-150

-100

-50

0

50

100

Dep

th [

m]

Analytical results using Snell's law

SWAN Higher Order Scheme, DX = 2000m

SWAN BSBT Scheme, DX = 2000m

SWAN Higher Order Scheme, DX = 1000m

SWAN BSBT Scheme, DX = 1000m

Ardhuin & Herbers, 4-step, DX = 2000m

Depth[m]

Continental Shelf refraction (WWM)

0

5

10

15

20

25

30

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000Length [m]

Dire

ctio

nal s

prea

ding

[°]

-200

-150

-100

-50

0

50

100

Dep

th[m

]

Analytical results using Snell's law

WWM CRD-N1 scheme, DT = 600s, DX = 2000m

WWM CRD-N3 scheme, DT = 600s, DX = 2000m

WWM CRD-N1 scheme, DT = 150s, DX = 2000m

WWM CRD-N3 scheme, DT = 150s, DX = 2000m

Ardhuin & Herbers, 4-step, dx = 2000m

WWM CRD-FCT scheme, DT = 150s, DX = 2000m

Depth [m]

Continental Shelf refraction(WWM)

0

5

10

15

20

25

30

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Length [m]

Dire

ctio

nal s

prea

ding

[°]

-200

-150

-100

-50

0

50

100

Analytical results using Snell's law

WWM CRD-N1 scheme, DT = 150, DX = 1000m

WWM CRD-N3 scheme, DT = 150, DX = 1000m

WWM CRD-N scheme, DT = 150, DX = 1000m

WWM CRD-FCT scheme, DT = 150, DX = 1000m

Ardhuin & Herbers, 4-step, dx = 2000m

Depth

+ -• Implicit Iterative Direct Methods (SWAN)

– The whole equation is discretized at once.

– Lower order diffusive linear implicit schemes (BSBT) must be used in order

to maintain monotonicity

– Higher order schemes (e.g. S&L) are not monotone and will lead in the

vicinity of strong gradients in the solution to non physical results (

– A advantage is that the convergent solution is independed of the

integration time step.

– One deficiency is the strong numerical diffusion-

– Another deficiency is that the convergence of the whole scheme is difficult to prove (LAX) and showed in practice to be a problem of such an approach. One solution is under relaxation (e.g. Ferzinger & Peric) applied by Zijlema & Van der Westerhuysen for the SWAN model. However, under-relaxation increases usually the amount of iterations.

– The limiter acts also on the advective part! This is very bad in non-stationary situations … or somebody tell me why it should be good to limit the advective part

+ -• Operator Splitting Methods (OSM)

– Godunov theorem can be easily applied.

– Lax equivalence theorem can be easily proved in most cases.

– Decoupling of the certain time scales of importance and accounting for them numerically.

– Optimal numerical schemes can be used for the certain sub-problems, e.g. adaptive schemes.

– Source term integration can be done adaptive (Tolman) reducing the necessity of application of the limiter to the change in time.

– The global scheme converges to the analytical solution when the discretization scales approaches the infitesimal limits.

– Parallelization and vectorization can be done at the same time.

- Multi-scale, multi-code and multi-grid can be embedded.

Development of the WWM II

• The Residual Distribution Framework was successfully introduced in the WWM code and verified for different cases e.g.

– U.S. East Coast– Haringvliet Estuary– Continental shelf refraction (Ardhuin & Herbers)

• The WWM II was coupled with SHYFEM a SWE Finite Element Model. The coupled models SHYFEM/WWM was used for the simulation of the storm surge in the northern Adriatic and the Gulf of Mexico (Ferrarin et al., 2006)

• Implementation of the source terms according to Donelan et al., Babanin et al., Young et al. … • In the cue … 3-d wave – current coupling, Ardhuin et al.• Implementation of the WWM advection schemes in the WWIII by Florent, Fabrice and me

(done)• Parallelization using MPI (Florent & Fabrice) is (done)

– Multi-scale validation using SHOWEX, EPEL, YANGTZE, TAIWAN, NORTH SEA with Ardhuin et al.

• Multi-Grid approach using P1 (3 points, 2nd order space-time) and P2(6 points; 3rd space-time) elements

• Publish something … hopefully …

Questions?

• How is the diffuseness of the numerical schemes included in the parameterizations of the physics and how about the conclusions from numerical results with respect to real physics?

• How can we interpret physics in numerical results with no or very little knowledge about the numerical schemes and their errors when we are using them?

• Possible solutions for the outlined problems of OSM are discussed in the next part of the lecture.

OSM in multi-physic applications

Decomposition methods are a powerful method of numerical investigation of complex (physical) time-dependent models, where the stationary part (elliptic) part consists of simpler operators, e.g.:

• Transport-Reaction Processes, see [Geiser, 2006; Hundsdorfer, Verwer, 2003] (Physical splitting)

• Hamiltonian Systems, see [McLachlan, 1994; Hairer, Lubich, Wanner, 2002] (Symplectic splitting)

• Air pollutant models, see [Zlatev, 1995] (Operator Splitting)• Maxwell equations, see [Horvath, 2006] (Operator Splitting)• Wave Action Equation (e.g. Tolman, 1992 and many others )

Attributes of OSM

Available decomposition methods:

• Time-decomposition methods• Spatial-decomposition methods

Contribution of the decomposition of the whole equation

• Decoupling the time-scales, space-scales (Reduce the stiffness in single operators)

• Decoupling the multi-physics. (Reduce the unphysical behavior with best choice of discretization and solver methods, e.g. CFL-conditions)

• Time-adaptivity, Space-adaptivity. (Efficiency and accuracy in computations).

• Parallelization in Time and Space. (Reduction of computational time).

• Results : More efficient and fast algorithms with high accuracy, simple implementable.

Basic Idea

• Idea : Decoupling into simpler parts with respect to operators, dimensions, times, models

• Effect : More efficient computations, parallelization, adequate discretization and solver methods for each part

Deficiencies: • Decoupling of the global time step – unphysical results.• Order reduction in the stiff case

Splitting Error and Comutativity

***

** ** 0 x y

Nc N c N

t x y

***** ** 0 x y

Nc N c N

t x y

c• The X and Y operator compute on a FDM mesh and so we have no splitting error.

• This is why the splitted Ultimate Quickest scheme is better then the un-splitt counterpart maintaining the mixed terms due to the 2-d discretization

• Since theta and sigma are orthogonal operators there is no splitting error when splitting the intra-spectral propagation. However, care must be taken in the stiff case.

***

** ** 0 x y

Nc N c N

t x y

xc yc

Problems of the Classical Splitting Methods

Error analysis• Error is of first order global in time :• Local and global error for the decomposition and the full

solution

1 1

( ) 2

( ) ( )

1( )

2

; ; ;

n nnum exactN

N n

x x x x tot

e N t N t

BA AB CA AC DA ADeN t O

CB BC DB BD DC CD

A c c B c C c D S

• Here is the global error and the error describes the situation if non of the terms is comuting otherwise one gets exact solution (Strang, 1968)

)( 2O 1nt t

Physical Error

• In each equation, we solve a different problem due to the full equation, e.g. no influence of the intermediate time-steps to the other equations.

• We only interact at the initial conditions, so the time-scales of the operators A and B are independent and therefore not interacting as in the original equation, e.g. coupled transport equations [Geiser 07].

1 1 1

12 2 2 1

,

,

n nt

n nt

N AN with N N

N AN with N N

Iterative Operator Splitting Methods

Benefits:• Larger time-steps are possible in each iterative step• Spectrum of the operator allow to control the stiffness of the

operators• Higher order can be claimed with more iterative steps• Efficient and simple implementation of the iterative schemes• Parallel algorithms can be used

Application to the Wave Action Equation

We solve iteratively four equations:

where i = 1,2,3…, N0 = 0, Nin = Nn

1 1 1,

1 1 1 1,

3 1 2 2,

t i x x i x y i i i i tot

t i x x i x y i i i i tot

t i x x i x y i i i i tot

N c N c N c N c N S

N c N c N c N c N S

N c N c N c N c N S

errNN ni

ni

1

11

Analysis: Iterative Operator Splitting Methods

• We concentrate on two operators:

Where N0(t) is any fixed function for each iteration. (Here, as before, un denotes the

known split approximation at the time level t = tn) The split approximation at the time-

level t = tn+1 is defined as Nsp,n+1 = N2m+1(tn+1).

(Clearly, the functions Nk(t) (k = i−1, i, i + 1) depend on the interval [tn, tn+1], too, but,

for the sake of simplicity, in our notation we omit the dependence on n)

1

11 1

, N

, N

i n ni i i

i n ni i i

N tAN t BN t with t N

tN t

AN t BN t with t Nt

Decomposition based on operator spectrum

• To detect the operators in the differential equation as stiff or non-stiff operators, we can apply the Eigen values of each operator and use them as reciprocal time scales. The operator equations are analyzed with the Eigen value problem:

• where the operators A and B result form the spatial discretization. • The Eigen values are detected in the decoupled equations:

• Based on the Eigen values λA, λB we can propose the time steps ΔtA=1/ λA

and ΔtB=1/ λB

1

0

, ,n nt A B

n

c t A B N t N t t t t

N t N t

10

10

, , ,

, , ,

n n nt A

n n nt B

N t AN t N t t t t N t N t

N t BN t N t t t t N t N t

Decomposition MethodsError for the Iterative splitting-method

• Theorem: The error for the splitting methods is given as :

122012

21

mn

mnmm

nini

OeKe

henceand

OeBKe

where n is the time-step, e0 the initial error e0(t) = N(t) − N0(t) and m the number of iteration-steps, K and Km are constants, ||B|| is the maximum norm of operator B and A, B are bounded, monotone operators. The error can be controlled by the operator B, e.g. non stiff operator.Proof : Taylor-expansion and estimation of exp-functions. Seethe work Geiser,Farago (2005).

Parallelization of the Time-Decomposition method: Windowing

• The idea for parallelization in time are the windowing, that the processors has an amount of time-steps to compute and to share the end-result of the computation as an initial-condition for the next processor.

Outlook

• Experiments for the Wave Action Equation• Theoretical Investigations in multiple iterative operator

splitting methods• Parallel implementation

On perspectives for the numericalSolution of the WAE: Experiments andDecomposition Methods

Dipl.-Ing. A. RolandDr. GeiserProf. Dr.-Ing. habil Prof. h.c. U.C.E. ZankeProf. Dr. S-H. OuProf. Dr. T-W. HsuDr. J-M. Liau

Jaws, HawaiiBjörn Dunkerbeck, E11World Champion

Mast length = 4.2m