Transcript Document
-SELFE Users Trainng Course (2) SELFE numerical formulation
Joseph Zhang, CMOP, OHSU
1
Numerical methods for pde
1!
(
f
2!
( ) 2
f n
!
( )
n
f
(
n
1) (
n
1)!
( )
n
1 2 System of algebraic equations Truncation error consistency, and order of convergence (Lax) Stability Higher-order approximation Upwind
c j n
1
c j n
D
t
u c n j
D
x c j n
1
c j n
1
c n j
u
j n
c j n
1 1 j-1
u
j Finite element Residual Weighting function Galerkin FE Least square j+1 D
x
0
L u
ˆ ,
w
wd
0
w
w
Basis/shape function (satisfied at finite number of points) Positive definite operator positive definite matrix Symmetric mass matrix
N x
Strong vs. weak formulation
Shape function
Used to approximate the unknown function Although usually used within each individual elements, shape function is global not local!
u
i N
1
u i
i
,
i
(
x j
)
ij
Must be sufficiently smooth to allow integration by part in weak formulation Mapping between local and global coordinates Assembly of global matrix 3
L
i N
1
u
i i
j d
M
m
1
m L
N
i
1
u
i i
j d
m
0,
j
1,...,
N
j Conformal Non-conformal 1 1 j-1 j j+1 j-1 j j+1
x N x N
Semi-implicit Eulerian-Lagrangian Finite Element (SELFE)
A formal semi-implicit finite-element framework A key step is to decouple the continuity and momentum equations through the bottom boundary layer Unstructured grid in the horizontal Hybrid SZ in the vertical Eulerian-Lagrangian method (ELM) for momentum advection Volume conservation is not enforced numerically (but good) Numerical efficiency: semi-implicit time stepping; ELM Mild stability constraint: horizontal viscosity, baroclinic pressure Moderately more expensive than ELCIRC Convergence rate: 2 nd order for uniform grid Optional higher-order numerics for momentum and transport Treats inundation/drying in a natural way (validated for inundation benchmarks) Bed deformation scheme incorporated (e.g., tsunami) 4
Horizontal grid
(
i,j
) 5
k z
Vertical grid (1)
Pure S zone
N z
s zone
h c
S zone
h s
SZ zone
z
s =0
h s S
-levels s =-1
k z
1
C h
2 1
Z
-levels
s
)
z
(1 s )
h c
s (
h
b
) sinh( s sinh
f
)
b
tanh
f
s
c
1/ 2) 2 tanh(
f
tanh(
f
/ 2) / 2) s 0) (0
b
1; 0
f
20) 6
S
-coordinates
f =10 -3 , b =1 f =10, b =0 f =10, b =1
h c
f =10, b =0.7
h s
7
z
=
h s
Vertical grid (2)
8
Vertical grid (3)
9 • 3D computationa unit: uneven prisms •
S
-coordinates (Song and Haidvogel 1994) are ideal for shallow region • •
Z
-coordinates are necessary to “stabilize” the deep region s -coordinates are used where
S
-coordinates are invalid • Equations are not transformed into
S
- or s -coordinates, but solved in their original (
Z
) form • Interpolation mode: along • Hydrostatic consistency:
Z
or
S
(in pure
S
region) • pressure Jacobian with higher-order integration •
Z
-method (preferred)
P
ˆ
n
k
+1
k
+1
S
ˆ
k z
2
s
2
z
1
s
1
Semi-implicit scheme
10 Continuity Momentum b.c.
D
u
Dt
n
1 D
t
n
h
u
n
1
dz
)
h
u
n dz
0
u
n
1
u
* D
t
f
n
n
1
g
(1
n n
u
z
u
n
1
n
1
z
τ
n
1 , at
z w
n
u
b n
1 , at
z
n
;
h
,
n
+
z
n
n
u
n
1
z
C D
|
u
b n
|
u
n
1 D
t
u
* ;
u
*
t t
) dispersion diffusion Implicit treatment of divergence and pressure gradient terms by-passes most severe CFL condition ELM: takes advantage of both Lagrangian and Eulerian methods Grid is fixed in time, and time step is not limited by Courant number condition Advections are evaluated by following a particle that starts at certain point at time t and ends right at a pre-given point at time t+ D t.
D
x
The process of finding the starting point of the path (foot of characteristic line) is called backtracking, which is done by integrating d x/ dt =u 3 backward in time.
To better capture the particle movement, the backward integration is often carried out in small sub-time steps ( D t / N).
Simple backward Euler method as an option 5th-order embedded R-K method as an alternative Interpolation-ELM does not conserve mass; integration ELM does Numerical diffusion vs. dispersion Other considerations • Implicitness factor 0.5 ≤ ≤1 to ensure stability • Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, part of bottom velocity
t
+ D
t
Characteristic line
t
1
Finite-element formulation (1)
A Galerkin weighted residual statement for the continuity equation:
i
n
1 D
t
n d
i
U
n
1
d
v
i U
ˆ
n n
1
d
(1 )
i
i
U
n d
i n U d v
+
U
u
dz
h
Vertical integration of the momentum equation:
v
v
i U n n
1
d
0, (
i
1,...,
N p
; 0.5
1)
U
n
1
G
n
H n n
n
1
n
D
t
u
n b
1
C D
|
u
b n
|;
G
: explicit terms
u
b
: bottom velocity Momentum equation applied to the bottom boundary layer:
u
b n
1 D
t
u
*
b
f
b n
n
1
g
(1
n
+
z
n
u
n
1
z
, at
z
b
h
via
u
z
0 ln(
b
/
z
0 )
C
1/ 2
D
|
u
b
|
u
b
, (
z
0
z
b
h
)
b
u
b
11 z = h
Finite-element formulation (cont’d)
Vertically integrated velocity becomes:
U
n
1
n
n
H n
n
D
t n
n
1 , Finally, substituting this eq. back to continuity eq. we get one equation for elevations alone: I 1 I 2 I 3
i n
1
g
2 D 2 ˆ
n
i n
1
d
g
2 D
t
2
v
i n
n
1
n d v
t
v
i U
ˆ
n n
1
d
v I n
12
I n
i n
)
t
I 4
i
U
n
i n
d
)
t
i n U d n
I 5
t
v
i
I 6
n d
v
Notes: • When node i is on boundaries where essential b.c. is imposed, equations are needed, and so I 2 • When node i the last term is known and I 6 only the first term is truly unknown!
is known and so no need not be evaluated there is on boundaries where natural b.c. is imposed, the velocity is known and so
Matrices: I
1
I
1
N i j
j
(
i
'
n
1
g
2 D ˆ ˆ
i
'
i
’
j N i
1
l
3 1
n
1 1 12
A j
g
2 D
t
4
A j
is local index of node I in j .
n
1 )
d
j
reaches its max. when
i
’=
l
, and so does I 1 .
i
3
i
' 0 so diagonal is dominant if the averaged friction-reduced depth ≥0 (can be relaxed) Mass matrix is positive definite and symmetric!
(
i’,1
)
i
’
i
> ' i j (
i’,2
) 13
Matrices: I
3
and I
5
I
3
v
i U
ˆ
n n
1
d
v
j
ij
i U
ˆ
n n
1
d
ij
j L
4
ij N z
1
bs
D
z
1 (
u
ˆ n
n
1 ) 1 (
u
ˆ n
n
1 ) Flather b.c. (overbar denotes mean)
u
ˆ n
n
1
u
n
I
3
j L ij
(
U
n 2 )
ij
n
1 )
gh ij
6 2(
i n
1
i
) (
j n
1
j
) j ij
n
i j Similarly
I
5
j
ij
ˆ ˆ
i
'
U d n n
ij
j L
4
ij N z
1
bs
D
z
1 (
u
ˆ n
n
) 1 (
u
ˆ n
n
) 14
Matrices: I
4 Has most complex form 15
I
3
i n
'
j
j
)
t
i
U
n
t
i d
j
j
h
u
*
dzd
D
j t
j
F
1
d
d
N i
j
A j
12
l
3 1
n
(1 ) (1 )
tA j
ˆ
i
'
U
n j j A t j
F
2
τ
w
(
u
*
b
D
tf
2
b
D
tf
1
b
)
g
(1 )
i
G
'
j
n
We have decomposed vector f into two parts: one for averaging and the other for integration
f f
1
f
2 Most complex part is the baroclinicity
f
c
g
0
z
at elements/sides
Baroclinicity: pressure Jacobian method
• No longer used in the newest version • In s S layers
z
s s s s
z
is evaluated using cubic spline in the vertical In Z layers, the derivative can be evaluated directly The vertical integration is done using higher-order rule (Song and Haidvogel) Advantage: no special treatment is needed near surface and bottom Disadvantage: pressure gradient errors 16
Baroclinicity:
Z
-coordinate method (preferred)
Interpolate density along horizontal planes
z
Advantage: alleviate pressure gradient errors (Fortunato and Baptista 1996) Disadvantage: near surface or bottom Method: compute derivatives along two direction at node then convert them back to x,y .
i and
ELM transport
• Density is defined at nodes and whole levels • Density gradient calculated at side centers and whole levels • Either constant extrapolation (shallow) or more conservative method (deep) is used near bottom • Cubic spline is used in all interpolation of (and necessary S,T) ( ) • Solve 2 eqs for the density gradient (
x
' ) (
x
2
x
1 ) (
y
' ) (
y
2
y
1 ) ' 2 1 ' (
x
' ) (
x
4
x
3 ) (
y
' ) (
y
4
y
3 ) 4 ' 3 ' Horizontal boundary or dry interface: no flux b.c.
3 ( j,k ) 2 ( j,k ) 1 17 4
Baroclinicity: upwind/TVD transport
• Density is defined at prism centers (half levels) • Density gradient calculated at prism centers first (for continuity eq.); the values at face centers are averaged from those at prism centers (for momentum eq.) • Either constant extrapolation (shallow) or more conservative method (deep) is used near bottom to avoid spurious flow • Cubic spline is used in all interpolation of averaging for sides for 3 pairs; e.g.
( ) except for the • 3 eqs for the density gradient vs. 2 unknowns – averaging needed 1 ( ' ) (
x
1
x
0 ) ( ' ) (
y
1
y
0 ) 1 ' 0 ( ' ) (
x
2
x
0 ) ( ' ) (
y
2
y
0 ) 2 ' 0 ' • degenerate cases • Vertical integration using trapzoidal rule i 0 2 18 3
Horizontal viscosity
Evaluate it at 3 side centers, and integrate using linear quadrature
u
)
u
n d
d
l
1
A l
l
3
i
1
s
u
n
The normal derivative is computed using average between adjacent elements
u
n
l
u
n
l
Use linear shape function within each element and chain rule to calculate the derivatives
u
n
u
n
s s side j 19
Momentum equation
b
h
l
(
l
( u,v ) solved at side centers and whole levels • Easy of imposing b.c.
• Staggering for stability Galerkin FE in the vertical
k b
u
n
1 1, D
t
z
N z
)
u
n
1
z
dz
b
h
l
u
*
n
1 D
t
f
n
1
g
(1
g
Solution at bottom from the b.c. there (u=0 for ≠0)
D
u
Dt
f
+
z
u
z
u
z
u
z
τ
w
u
b
, at
z
h
n
dz
( A (
z N z
D
t
τ
n w l
) 1 D
z l
6 1 + A ( (2
u
N z l n
1
l
)
u
l n
1 1 D
z l
1 6 ) (2
g
l l
1/ 2 D
t
g
l
1 )
u
l n
1 1 D
z l
u
1
l n
1 A (
k b
A (
l
) D
z l
6
k b
(2
g
l n
1
l
) D
z l
6
g
l n
1 1 ) (2
u
l n
1
l
b
1,
N z
)
u
l n
1 1 )
l
1/ 2 D
t u l n
1 D
z l u l n
1 1
b
1 D
t
n
u
k b
1 1 The matrix is tri-diagonal z = h b 20 N z k b +2 k b +1 k b
Velocity at nodes
Needed for ELM (interpolation) • Method 1: averaging around ball (most diffusive) • Method 2: use linear shape function (conformal or non conformal); optional averaging 3
u
1
u
II
u
III
u
I
II • Filtering of modes • Needs velocity b.c.
1 III I • Linear interpolation at foot of char. Line • Conformal • Non-conformal – discontinuity along each side 2
i
21
Shaprio filter
Few ocean models are truly free from spurious numerical modes (myriads of processes) A simple filter to reduce sub-grid scale (unphysical) oscillations while leaving the physical signals intact ( a =0.5 optimal) 0
u
0 a 4
i
4 1
u
i
4
u
0 , (0 0.5) No filtering for boundary sides – need b.c. there especially for incoming flow Pre-processor to check the geometry (ipre=1 with 1 processor) Violations usually occur at boundary (fort.11) 1 2 0 3 4 22 internal boundary extreme
ELM with Kriging
• Best linear unbiased estimator for a random function • “Exact” interpolator • Works well on unstructured grid f (x) a a 1 2
x
a 3
y
i N
1
i K
(|
x
x
i
|) Drift function Fluctuation K is called generalized covariance function Minimizing the variance of the fluctuation we get
i N
1
i
0
i N
1
i x i
0
i N
1
i y i
0
f x y i
)
f i
2
h h
3 x 23 2-tier Kriging cloud
Vertical velocity
Vertical velocity using Finite Volume
S
ˆ
k
1 (
u k n
1 1
n k x
1
v k n
1 1
n k y
1
w n
1 1
n k z
1 ) ˆ
k m
3 1
q
ˆ
j k
u
s
n
j
(
q
ˆ
n
1
q
ˆ
n
1
k n
1
n k x
v k n
1
n k y
w n
1
n k z
) 1 ) / 2 0, (
k
k b
,...,
N z
1) 24
w n
1 1 (
k
k b k
1 1 ˆ
k
1 ,...,
N z
1)
m
3 1
s
(
q
ˆ
n
1
q
ˆ
n
1 1 ) / 2
S
ˆ
k
1 (
u k n
1 1
n k x
1
v k n
1 1
n k y
1 ) ˆ
k k n
1
n k x
v k n
1
n k y
w n
1
n k z
) , Bottom b.c.
u k n
1
n k x
v k n
1
n k y
w n
1
n k z
b
t
, (
k
k b
)
P
ˆ
n
k
+1
k
+1
S
ˆ
k
Inundation algorithms
Algorithm 1 Update wet and dry elements, sides, and nodes at the end of each time step based on the newly computed elevations Algorithm 2 (accurate inundation for wetting and drying with sufficient grid resolution) Wet Dry add elements & extrapolate velocity remove elements Iterate until the new interface is found at step n+1 Extrapolate elevation at final interface (smoothing effects) dry 25 A n B wet Extrapolation of surface
Transport: ELM
Galerkin FE applied to both nodes and sides • no horizontal diffusion
h
l T n
1 D
t T
*
dz
h
l
z
T
z n
1 Apply mass lumping reduces FE to FD
n
Dc Dt
c
z c
z
n
c
z
Q c
,
z
0,
z
(
h
c
),
c
h
26 D
z l
1/ 2
T l n
1
l
1/ 2 D
t n
1
T l
1
T l n
1 D
z l
1
l
1/ 2 D
t T l n
1
T l n
1 1 D
z l
D
z l
1/ 2 (
T l
*
Q l n
D
t
), (
l
b
1, D
z l
2
T l n
1
N
1/ 2 D
t T l n
1
T l n
1 1 D
z l
D
z l
2
T l n
1
l
1/ 2 D
t T l n
1 1
T l n
1 D
z l
1
T
ˆ
t
D
z l
2 (
T l
*
Q l n
D
t
),
l
N z
D
z l
2 (
T l
*
Q l n
D
t
),
l
k b
ELM: mass conservation not guaranteed • Linear with element splitting in the horizontal; cubic spline in the vertical • Quadratic in 3D (works only in estuary due to dispersion) • Impose bounds from surrounding nodes to alleviate dispersion • Kriging (coming up) • Vertical interpolation: placement of T and • Upwind bias for quadratic interpolation Char. line ,
N z
1)
T l
x
l l
+1
l
-1
l
-2
Transport: upwind (1)
T
t
T
z T
n
(
u
T
)
z
T
z
(
h
T
)
T
ˆ,
z
0,
z
h
Finite volume discretization in a prism ( i , k ): mass conservation
V i T i m
1 D
t
'
T i m
5
j
V u S T j j j
*
V Q i m
D
t
' 3
j
1 (
h
)
j T j m
ij T i m S j
, (
k
k b
1, D
i
' ,
N z
)
T m
1 1 D
z
T m
1 1/ 2 1
T m
1 D
z
T
1/ 2
m
1 1 S j S S k T i,k k 1 m≠n, D t’ ≠ T i = T i,k; D t; u j is outward normal velocity 27 Focus on the advection first
T i m
1
T i m
D
t V i n
'
j
V j
* From continuity equation
Q j
u S j j
Q
|
Q
| |
Q
|
j j j
j
S + : outflow faces; S : inflow faces Upwinding
T j
*
T T i j
,
j
,
j
S
S
Transport: upwind (2)
Drop “m” for brevity
T i m
1 (1 a
T i
)
T i
D
t
'
V i n
a
T j
|
j i
|
j j
1 Max. principle is guaranteed if D
t
'
V i
|
Q j
|
T i
D
t
'
V i
|
j j
S j S S T i,k k 1 1 D
t
'
V i
|
Q j
|
t
'
V i
|
Q j
| Courant number condition (3D case) Global time step for all prisms Implicit treatment of vertical advective fluxes to bypass vertical Courant number restriction in shallow depths
V i T i m
1 D
t
'
T i m
k
2 1
u S T k k k m
1 *
j
3 1
u S T V Q j j j
*
i n
ˆ
n
D
i
'
T m
1 1 D
z
T m
1 1/ 2 1
T m
1 D
z
T m
1 1 1/ 2 D
t
'
j
3 1 (
h
)
j T j m
ij T i m S j
V Q i m
, (
k
k b
1, ,
N z
) (Modified Courant number condition) Mass conservation • Budget – vertical and horizontal sums • movement of free surface: violation of volume conservation k 28
Solar radiation
Q
ˆ 1 0
C p
H
z
total solar radiation
H
(0) Re / 1 2 Budget 0
H
(0) 0
C p
The source term in the upwind scheme (F.D.)
i i m
V i
0
C p H m b
0
H m
D
z H m
1
i m
H m
1 ) 0
C p
to ensure total heat is conserved (black body) This can cause overheating in shallow depths: adjust albedo 29
Transport: TVD (1)
Start from discretized advection Now include downwind component '
T j
* Flux limiter function (depends only on face upwind j ); Downwind and central schemes; The key is to find an appropriate function that does not violate max principle
T i m T j
* ' D
t V i n
2
j
'
T
j
V jD j
*
T j
* upwind 0
j
2
T i m
1
T i m
D
t V i
j
S
|
Q j
| (
T j T i m
1
T i
) D
t V i
j
S Q j
2
j
(
T jD
T j
* )
T i m
D
t V i
'
j
S
|
Q j
| 1 2
j
(
T j
T i
) D
t V i
'
j
S
|
Q j
|
j
2 (
T i
T j
) Sum of coefficients = 1; so max principle is satisfied if and only if the coefficient of negative T i is non Put the last term in similar form as the upwind term D
t V i
'
j
S
|
Q j
| 2
j
(
T i
T j
) D
t V i
'
j
S
|
Q j
|
j
2 (
T i
T j
)
j
S
|
Q j
| (
T j
T i
) 30 D
V i t
'
j
S
|
Q j
|
j
2 (
T i
T j
) D
t
'
V i
j
S
|
Q j
| (
T j
T i
)
Transport: TVD (2)
T i m
1
T i T i
D
V i t
' |
Q j
| 1
j
2 1 D
V i t
' |
Q j
| 1
j
2 (
T j
T i
) D
t
'
V i
|
Q j
|
j
2 (
T j
T i
) D
t
'
V i
|
Q j
| 1
j
2
T j
Max principle is satisfied if (Courant number condition) 1 |
Q j
|
j
2
T i
T j
|
Q j
|
j
2 |
Q m
T i
|
T m
T j
T i
j
2
r j r j
Since |
Q m
|
Q j
| |
T m
T i
T j
T i
Upwind ratio r 0
j
2, 0
j r j
2, 0
N
The Courant number condition is
t V i
(1
N
) |
Q j
| as compared to upwind
t V i
|
Q j
| D
t
'
V i
1
j
2 |
Q j
| 1
j
2 0 0 j S + i S m 31
Transport: TVD (3)
TVD scheme • Mass conservative with max principle; second-order accuracy • Fully explicit scheme due to nonlinearity – vertical Courant number most severe • Boundaries and wet/dry interface – revert back to upwind • Other higher-order schemes: MUSCL; WENO Choice of flux limiting function (gradient detector) max(0, min(1, 2 ), min(2, ))
r
Super Bee (compressive) Minmod (diffusive) Osher Van Leer 32 Second order TVD region (Sweby 1984)
Turbulence closure (1)
Dk Dt
z
k
k
z
M
2
N
2
D
z
z
c
2
c
N
2
k l
1 (
c
0 0 ) 2 D ,
z
(
c
0 )
p k
Essential b.c.
33
u
z
h
or
m
( ,
z
0 D )
n
h
,
z
or
h
or
wall
and
k
k
z
z
z
Natural b.c.
0,
z
h
or 0
n
0
n
,
z l
l
,
z
h
A (
l
N z
) D
z l
1 6 2
k l n
1
k l n
1 1 2
k l n
A (
l
k b
) D
z l
6 2
k l n
1
k l n
1 1 2
k l n
k l n
1 (
k
)
l
1/ 2 D
t k l n
1 1 D
z l k
1
l n
1
k l n
1 (
k
)
l
1/ 2 D
t k l n
1 D
z l k l n
1 1 (
b
, , A(
l
N z
) D
t
D
z l
1 6
k l n
1/ 2 D
z l
1 2
M
2
M
2
N
2
N
2
l n
1/ 2
l
n
1/ 2 (2
k l n
1 A (
l
k b
) D
t
D
z l
6
k l n
1/ 2 D
z l
2
M
2
M
2
N
2
N
2
l n
1/ 2
l
n
1/ 2 (2
k l n
1
k l n
1 1 ) (
c
0 ) 3
k
1/ 2
l
1
n l
1/ 2
k l n
1 1 ) (
c
0 ) 3
k
1/ 2
l
1
l n
1/ 2 D
z l
6 D
z l
1 (2
k l n
1 6
k l n
1 1 ) (2
k l n
1
k l n
1 1 )
N z
)
Turbulence closure (2)
A (
l
A (
l
N z
) D
z l
6 1
k b
) D
z l
6 2
l
2
l n
1
n
1
l n
1 1
l n
1 1 2
l n
2
l n
l n
1
l n
1 ( ( )
l
1/ 2 D
t
l n
1 1 D
z l
1
l n
1 )
l
1/ 2 D
t
l n
1
l n
1 1 D
z l
A(
l
N z
) D
t
(
c
6 D 0 )
z l
3 D
k l n
1/ 2 1
z
2
l c
1 2
c
k
c
1 1/ 2 1
M M
2 2
l F wall
c
c
3
l n
1/ 2 3
N
2
N
l
D
z l
6 2
n
1
l n
1/ 2 1/ 2 (2 (2
l n
1
l n
1
l n
1/ 2
l n
1 1
l n
1 1 ) ) A (
l
k b
) D
t
(
c
D
z l
D 6
k l n
1/ 2 0 ) 3 2
c z
l
2
c
k c
1 1/ 2 1
M M
2 2
l F wall c
c
3 3
n l
1/ 2
N N
2 6 2 D
z l l n
l n
1/ 2 1/ 2 (2 (
l
k l n
1/ 2 2
n
1
l n
1
l n
1 1 )
l n
1 1 ) (
l
k b
, ,
N z
) 34
GOTM turbulence library
General Ocean Turbulence Model One-dimensional water column model for marine and limnological applications Coupled to a choice of traditional as well as state-of-the-art parameterizations for vertical turbulent mixing (including KPP) Some perplexing problems on some platforms – robustness?
Bugs page Division by 0?
35 www.gotm.net
Numerical stability
Semi-implicitness circumvents CFL (most severe) (Casulli and Cattani 1994) ELM bypasses Courant number condition for advection Implicit transport scheme along the vertical bypasses Courant number condition Explicit terms Baroclinicity internal Courant number restriction Horizontal viscosity/diffusivity D
x
diffusion number condition Upwind and TVD transport – Courant number condition Coriolis – stable but prone to “modes” 1 D
t
D
x
2 0.5
3 2
u
1
n
1
u
2
n
1
u
3
n
1 0 36 1 2
Lateral b.c. and nudging
37 S&T u,v Variable Type 1 (*.th) Time history; uniform across bnd Type 2 (param.in); Const.
Type 3 (param.in); tides Type 4 (*3D.th); Time history Type -1 (param.in); radiation Type -4 (*3D.th); nudging Nudging near bnd Via discharge Via discharge Clamp at i.c.
Tides (uniform across bnd) Nudge to i.c.
Nudge to *3D.th
Flather Nudge to uv3D.th (2 separate relaxation for in & outflow)
Summary
Time stepping Horizontal grid Vertical grid Numerical algorithm Convergence rate for non orthogonal grid Advection (momentum) Advection (transport) Volume conservation Wetting/drying ELCIRC Semi-implicit Orthogonal unstructured Z -coordinates Finite difference / finite volume Divergence ELM ELM/upwind Enforced Yes SELFE Semi-implicit unstructured Hybrid SZ coordinates Finite element / finite volume At least 1 st ELM (optional higher order Kriging) ELM/upwind/TVD Not enforced (but good) Yes 38