Modeling & Computation of Sea Surface Heights in Complex Domains: Techniques, Slides of Computational Techniques

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Modeling and Computation of Sea Surface

Heights in Complex Domains

2

Shallow Water Equations

•^

Good approximation to fluid motion equations– When fluid density is homogenous and depth is small in

comparison to characteristic horizontal distances

• Coastal areas– Reduces complicated 3D problem to 2D -^

In solving 3D primitive hydrostatic equations withtop surface of the fluid free to move– Free surface allows gravity wave propagation at speed– Gravity wave speed greatly exceeds advective velocity

in deep of oceans and is culprit of restrictions on timesteps

gh

4

Time Splitting for Implicit Terms Only

•^

Uzawa-like splitting used for pressure and velocityunknowns. Pressure equation results in a SPD system– Can use an inexpensive, robust CG solver– Matrix never stored– Only a matrix-vector operation available

-^

Preconditioners– Standard is a diagonal based one– New one specifically tailored to Spectral Ocean Element

Model (SEOM) using an Additive Schwarz Method (ASM)

5

Spectral Ocean Element Method (SEOM)

•^

Novel feature– Isopycnal coordinates vertically, spectral horizontally

-^

Benefits of spectral discretization– Geometric flexibility through unstructured and quasi-

unstructured grid

• Dual

h-p

paths to convergence

• Low numerical dispersion and dissipation errors– Dense computational kernels lead to splendid parallel

scaling

7

Multiple Layer Rationale

•^

Benefits– Ease of development (stacked shallow water equations)– Minimization of cross-isopycnal diffusion– Eliminates pressure gradient errors– Baroclinic process representation– Cost savings over full 3D model

-^

Appropriate for– Wind driven simulations, eddy formulations, and (in

part) flow/topology interaction

8

Derivation of Numerical Model

-^

is the velocity vector.

-^

is the sea surface displacement

-^

is the gravity acceleration

-^

is the resting depth of the fluid

-^

is a mass source/sink term

-^

is the generalized forcing term for the

momentum equations: Coriolis force, nonlinear advection,viscous dissipation, and wind forcing

-^

No slip boundary conditions are imposed to complete thesystem

^



and

(^

)^

.

t^

t

u

g

F

h^

u^

Q







  







^



^ g

( , ) u^

u v  ^ h Q

(^

,^

)

x^

y

F^

f^

f

 

10

Variational Form

In Cartesian coordinates:where

and

are the velocity and pressure test

functions. The integration by parts leads to a SPDsystem.

, x

t^

x

A^

A^

A

u dA

g^

dA

f^

dA



 



^



^

^



, y

t^

y

A^

A^

A

v dA

g^

dA

f^

dA



 



^



^

^



(^

)(^

)^

,

t^

x^

y

A^

A^

A

dA

u^

v^

h^

dA

Q^

dA



^

^





^

^

^



^

^



11

Semi-Implicit Integration

•^

Gravity waves– Terms

and

cause trouble.

• Use Crank-Nicolson scheme to solve implicitly. -^

Rest of the terms– Do not cause trouble.– Use third order Adams-Bashforth scheme to solve

explicitly.

g^ 

(^

) hu  



13

Spatial Discretization

•^

Spectral formulation with N

p^

= N

v^

velocity:

pressure:

•^

For each element, we can write

, ,^1

, ,^

1

( ,

)

( )

( ),

( ,

)

( )

( )

vN v

v^

v

k l^

k^

l

k l N

p^

p

i j^

i^

j

i j

u^

u

^



^

 



  

^

^

 



   

 

^



14

Galerkin Formulation

•^

Reduces problem to algebraic equations with a, b,c obvious substitution. Let

and

be the

mass matrices,

and

be the discrete gradient

operators, and

and

are components of the

discrete gradient operators. Then the Galerkinformulation is simply

1 1

1

.5.

.

. v^

x

v^

y

x^

y^

p

M u

gG

a

M v

gG

b

D u

D v

M

c









^



 



^



^



^

^

^



v M

p M

x G

y G

x D

y D

16

Matrix

S

Not Stored

•^

S

is

extremely

memory consuming.

• Many entries from nodes that are two layers of

elements away.

-^

S^

is never stored. The implemented implicit matrix- vector multiplication is fast and takes advantage of thelocal tensor product structure of the matrix.

17

Schoedinger-like Equation

•^

Schur complement system similar to Schoedinger-like equation

-^

S

symmetric, positive definite– Can use preconditioned conjugate gradient method– Preconditioner defined using matrix-vector multiply

lumped Schur complement.

^

^

^

^



1

1

1

N

N ii^

i^

i^

i

C

S

^



^



2

( )

1

( )

( )

( ),

4

( )

T g^

h x

x^

x^

f^

x^

x

m x

^

^

^





^



^





^

^

 

^



^



19

Local Element Contributions

•^

Consider only local contributions to generation oflocal matrices

, i.e., restrict the support of the

transformed pressure basis functions to theelements. Then

-^

Local matrices are similar to Schroedinger-likeequations in

r with homogeneous Neuman b.c.

^

^

^

^

^

^

^



^



nelem

T r

r=1 1

1

ˆ^

ˆ

4

r^

r

p^

T^

T

x^

v^

x^

y^

v^

y

r

r^

r^

r^

r^

r^

r

S^

A S A

M

g^

G

H

M

G

G

H

M

G





^



^



^



^



^



^





ˆSr

20

Calculating

Matrices

•^

Method 1: Derive

as a stiffness matrix of the

local Schroedinger-like equation

-^

Method 2: Use modified matrix-vector routinethat operates on a single element. Multiply all unitvectors to get columns of

for all local rows

i

• Consumes more CPU time, but is easy to do and robust.– We opted for Method 2. Time spent is a once per run

throwaway expense. After all, each run is

O(10,000)

to

O(1,000,000)

time steps.

ˆS

ˆS

, , ˆSr i j