1

-SELFE Users Trainng Course-(2)

SELFE numerical formulation

Joseph Zhang, CMOP, OHSU

2Numerical methods for pde

Finite difference Approximation of the differential operators using Taylor series System of algebraic equations Truncation error consistency, and order of convergence (Lax) Stability Higher-order approximation Upwind

Finite element

( ) ( 1)2 1'( ) ''( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1! 2! ! ( 1)!

( is between and )

n nn nf a f a f a f

f x f a x a x a x a x an n

a x

j j+1j-1

xx

1 N

u

11 1

1

n n n nj j j j n n n n

j j u j j

c c c cu c c C c c

t x

( )

ˆ( )

0

, 0

L u

L u

w wd

Residual

Weighting function

Galerkin FE w

(satisfied at finite number of points)

Positive definite operator positive definite matrix

Least square w Symmetric mass matrix

Basis/shape function

Strong vs. weak formulation

3Shape function Used to approximate the unknown function Although usually used within each individual elements, shape

function is global not local!

Must be sufficiently smooth to allow integration by part in weak formulation

Mapping between local and global coordinates Assembly of global matrix

j j+1j-1

x

1 N

j j+1j-1

x

1 N

1

, ( )N

i i i j iji

u u x

Conformal Non-conformal

1 1 1

0, 1,...,m

N M N

i i j i i j mi m i

L u d L u d j N

j

4Semi-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: 2nd 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)

5Horizontal grid

(i,j)

6

zone

Vertical grid (1)

1

hs

2

Nz

kz

=0

=-1

S-levels

Z-levels

kz1

z

(1 ) ( ) ( ) ( 1 0)

tanh ( 1/ 2) tanh( / 2)sinh( )( ) (1 ) (0 1; 0 20)

sinh 2 tanh( / 2)

min( , )

c c

f ffb b b f

f f

s

z h h h C

C

h h h

S zone SZ zone

Pure S zone

hshc

7S-coordinates

f=10-3, b=1 f=10, b=1

f=10, b=0 f=10, b=0.7

hc

hs

8Vertical grid (2)

z=hs

9Vertical grid (3)

• 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• -coordinates are used where S-coordinates are invalid• Equations are not transformed into S- or -coordinates, but solved in their original (Z) form• Interpolation mode: along Z or S (in pure S region)• Hydrostatic consistency:

• pressure Jacobian with higher-order integration• Z-method (preferred)

z2

z1s1

s2

zd

nk+1 k+1

kS

P

10Semi-implicit scheme

11 (1 ) 0

n nn n

h hdz dz

t

u u

1 11* (1 ) +

n nn n n ng g

t z z

u u uf

11

11

, at ;

, at ,

nn n n

w

nn n n

b

zz

z hz

uτ

uu

Continuity

Momentum

b.c. | |n nD bC u

1*

*; ( , , , , )nD

x y z t t tDt t

u uu

u

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+t. The process of finding the starting point of the path (foot of characteristic line) is called

backtracking, which is done by integrating dx/dt=u3 backward in time. To better capture the particle movement, the backward integration is often carried out in small sub-

time steps (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. dispersiont+t t

Characteristic line

1u t

x

diffusiondispersion

Other considerations• Implicitness factor 0.5 ≤≤1 to ensure stability• Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, part of bottom velocity

11Finite-element formulation (1) A Galerkin weighted residual statement for the continuity equation:

Vertical integration of the momentum equation:

Momentum equation applied to the bottom boundary layer:

11 1 1ˆ

(1 ) 0, ( 1,..., ; 0.5 1)

: shape functions; = + ; : impicitness factor;

v v

n nn n n

i i i n v i n v

n ni i n p

i v v

h

d d U d U dt

d U d i N

dz

U

U

U u

1 1 1

; | |; : explicit terms

: bottom velocity

n n n n n nb

n nD b

b

g H t t

H h C

U G u

u G

u

1 11* (1 ) + , at

n nn n n nb bb bg g z h

t z z

u u uf

b

ub

z=h1/ 20

00

| | , ( )ln( / ) D b b b

b

C z h z hz z

u

u uvia

12Finite-element formulation (cont’d) Vertically integrated velocity becomes:

Finally, substituting this eq. back to continuity eq. we get one equation for elevations alone:

1 1ˆ ˆ ,

ˆ

n n n n

n n n

g H t

H H t

U G

11 2 2 1 2 2 1ˆ ˆ ˆ

v v

nn n n n n n

i i i v i n vg t H d g t H d t U d In

ˆ ˆ(1 ) (1 )v

n n n n n ni i i i n i vI t t d t U d t d

U G n G

Notes:• When node i is on boundaries where essential b.c. is imposed, is known and so no equations are needed, and so I2 and I6 need not be evaluated there• When node i is on boundaries where natural b.c. is imposed, the velocity is known and so the last term is known only the first term is truly unknown!

I3I1

I4 I5 I6

I2

13Matrices: I1

'i

1 2 2 11 ' '

1

23',1

,1 1

ˆ ˆˆ( )

1 ˆ '12 4

i

j

i

Nn n

i i jj

Ni ln

j l jj l j

I g t H d

g tA Hi l

A

i

j

i’

(i’,1) (i’,2)>

reaches its max. when i’=l, and so does I1.'i li’ is local index of node I in j.

3

' 1

' 0i

i

so diagonal is dominant if the averaged friction-reduced depth ≥0 (can be relaxed)

Mass matrix is positive definite and symmetric!

14Matrices: I3 and I5

1 13

11 1

, 1 n , 1 n ,

ˆ ˆ

ˆ ˆ( ) ( )4

v ij

z

bs

n ni n v i n ij

j

Nij n n

ij k ij k ij kj k k

I U d U d

Lz u u

i

jij

jFlather b.c. (overbar denotes mean)

1 1n n

n 1 13

ˆ / ( )

( )2( ) ( )

2 6

n n

ijij n nij i i j j

j

u u g h

ghUI L

1

5 ' , 1 n , 1 n ,ˆ ˆ ˆ ˆ( ) ( )

4

z

bsij

Nijn n n

i n ij ij k ij k ij kj j k k

LI U d z u u

Similarly

n

15Matrices: I4

3'

3 , ', '1

ˆ ˆ ˆ ˆ(1 ) (1 ) (1 )12

iNjn n n n

i i i j l i l j i j i jj l

AI t t d tA t

U G U G

Has most complex form

'* 1 2 * 2 1

ˆ ˆ ˆ( ) (1 )j j j

nj j j j j w b b bh

d dzd t d A t tf tf g H

G G u F F τ u

We have decomposed vector f into two parts: one for averaging and the other for integration

1 2 f f f

Most complex part is the baroclinicity

0c z

gd

f at elements/sides

16

• No longer used in the newest version

• In S layers

Baroclinicity: pressure Jacobian method

zz

is evaluated using cubic spline in the vertical

In Z layers, the derivative can be evaluated directlyThe vertical integration is done using higher-order rule (Song and Haidvogel)

Advantage: no special treatment is needed near surface and bottomDisadvantage: pressure gradient errors

17

• 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)

• Mean density profile removed (optional)• Solve 2 eqs for the density gradient

Baroclinicity: Z-coordinate method (preferred)

Interpolate density along horizontal planes

Advantage: alleviate pressure gradient errors (Fortunato and Baptista 1996)Disadvantage: near surface or bottom

z

Method: compute derivatives along two direction at node i and then convert them back to x,y.

ELM transport

1

2

43

' ' ' ', 2 1 , 2 1 2 1

' ' ' ', 4 3 , 4 3 4 3

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x j k y j k

x j k y j k

x x y y

x x y y

(j,k)

(j,k)

( )z

Horizontal boundary or dry interface: no flux b.c.

18

• 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 except for the averaging for sides

• Mean density profile removed (optional)• 3 eqs for the density gradient vs. 2 unknowns – averaging needed for 3 pairs; e.g.

Baroclinicity: upwind/TVD transport

' ' ' ', 1 0 , 1 0 1 0

' ' ' ', 2 0 , 2 0 2 0

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x i k y i k

x i k y i k

x x y y

x x y y

( )z

1

2

3

0i

• degenerate cases • Vertical integration using trapzoidal rule

19Horizontal viscosity

3

, ,(side ),1 ( , ),

1( ) l i l ij k

l i l i kl

l

ud

n uu s

nd A

Evaluate it at 3 side centers, and integrate using linear quadrature

side j

Use linear shape function within each element and chain rule to calculate the derivatives

( ', )

side '

l j l

j

l

u

nu

n

The normal derivative is computed using average between adjacent elements

u u u z

n n z n

20Momentum equation

(u,v) solved at side centers and whole levels• Easy of imposing b.c.• Staggering for stability

+D

gDt z z

u u

f

at =w zz

u

τ

, at b z hz

u

uGalerkin FE in the vertical

1

1 1 1* (1 )

( 1, )

b b

nn n n n

l lh h

b z

t dz t g g dzz z

l k N

u

u u f

……

..

kb

kb+1

Nz

kb+2

z=-hb

Solution at bottom from the b.c. there (u=0 for ≠0)

1 1 1 11 1 1 1 11 1 1

1 1/ 2 1 1/ 2 , 1 11

1 11, 1

( ) (2 ) ( 1 ) (2 )6 6

( ) (2 ) ( 1 ) (26 6

b b

z

n n n nn n n n nl l l l l l

z l l l b l l l l k kl l

n nl ll N w z l l b l

z z u uN l t k l t t

z z

z zt N l k l

u uu u u u u

τ g g g

A A

+A A 11 )

( 1, )

nl

b zl k N

g

The matrix is tri-diagonal

g

21Velocity 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

• Filtering of modes• Needs velocity b.c.

• Linear interpolation at foot of char. Line• Conformal• Non-conformal – discontinuity along each side

i

1 2

3

III

III

1 II III I u u u u

22Shaprio filter

A simple filter to reduce sub-grid scale (unphysical) oscillations while leaving the physical signals intact (=0.5 optimal)

4

0 0 01

ˆ 4 , (0 0.5)4 i

i

u u u u

0

1

2 3

4

Few ocean models are truly free from spurious numerical modes (myriads of processes)

Pre-processor to check the geometry (ipre=1 with 1 processor)Violations usually occur at boundary (fort.11)

No filtering for boundary sides – need b.c. there especially for incoming flow

internal boundary extreme

23ELM with Kriging

• Best linear unbiased estimator for a random function f(x)• “Exact” interpolator• Works well on unstructured grid

x

2-tier Kriging cloud

1 2 31

( , ) (| |)N

i ii

f x y x y K

x x

Drift function Fluctuation

K is called generalized covariance function 2 3( ) , ln , ,K h h h h h Minimizing the variance of the fluctuation we get

1

1

1

0

0

0

( , )

N

ii

N

i ii

N

i ii

i i i

x

y

f x y f

24Vertical velocity

Vertical velocity using Finite Volume

nk+1 k+1

kS

P

1 1 1 1 1 11 1 1 1 1 , 1 1 ,

31 1

( , ) , ( , ), ( , ), 11

ˆ ˆ( ) ( )

ˆ ˆ ˆ( ) / 2 0, ( ,..., 1)

n x n y n z n x n y n zk k k k k i k k k k k k k i k k

n njs i m i m js i m k js i m k b z

m

S u n v n w n S u n v n w n

P s q q k k N

31 1 1 1 1 1 1 1

, 1 ( , ) , ( , ), ( , ), 1 1 1 1 1 1 ,11 1

1 ˆ ˆˆ ˆ ˆ( ) / 2 ( ) ( ) , ˆ

( ,..., 1)

n n n n x n y n x n y n zi k js i m i m js i m k js i m k k k k k k k k k k k i k kz

mk k

b z

w P s q q S u n v n S u n v n w nn S

k k N

, ,ˆ j k j k jq u n

Bottom b.c.

1 1 1, , ( )n x n y n z

k k k k i k k b

bu n v n w n k k

t

25Inundation 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 add elements & extrapolate velocity Dry remove elements Iterate until the new interface is found at step n+1 Extrapolate elevation at final interface (smoothing effects)

A

wet

dry

n

B

Extrapolation of surface

26Transport: ELM ( ), ( , )h

Dc cQ c c S T

Dt z z

, ˆ

0,

cc z

zc

z hn

1 * 1n nn

l lh h

T T Tdz Q dz

t z z

Galerkin FE applied to both nodes and sides• no horizontal diffusion

Apply mass lumping reduces FE to FD

l-1

l

l+1

Tl

1 1

1 1 1 11 *

1/ 2 1/ 2 1/ 2 1/ 21

( ), ( 1, , 1)l l l l

l

n n n nn n

l l l l l l b zl l

T T T Tz T t t z T Q t l k N

z z

1

1 11 *

1/ 2ˆ ( ),

2 2l l

l

n nn n nl l

N l l zl

T Tz zT t T t T Q t l N

z

1

1 11 *

1/ 21

( ), 2 2

l l

l

n nn nl l

l l l bl

T Tz zT t T Q t l k

z

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 interpolationChar. line

x

l-2

27Transport: upwind (1)

1 1 1 11 5, 1 , , , 1

* , , , 1, 1/ 2 , 1/ 2

3

1

''

' ( ) , ( 1, , )

m m m mm mi k i k i k i kmi i

i j j j i i k i i k i kj V i k i k

m mj i

h j j b zj ij

T T T TT TV u S T V Q A t

t z z

T Tt S k k N

k

k-1

Ti,k

3ˆ( ) ( )h

T TT Q T

t z z

u

ˆ,

0,

TT z

zT

z hn

S

SFinite volume discretization in a prism (i,k): mass conservation

Focus on the advection first

m≠n, t’ ≠ t;Ti=Ti,k;

uj is outward normal velocity

S

1*

'm mi i j jn

j Vi

tT T Q T

V

j j jQ u S

From continuity equation 0 | | | |j j jj j S j S

Q Q Q

S+: outflow faces; S : inflow faces

Upwinding *

,

, i

jj

T j ST

T j S

j

28Transport: upwind (2)

1 ' ' '| | | | 1 | | | |

(1 )

mi i j i j j j i j jn

j S j S j S j Si i i

i j

t t tT T Q T Q T Q T Q T

V V V

T T

k

k-1

Ti,k

S

S

SjDrop “m” for brevity

Max. principle is guaranteed if

'1 | | 0 '

| |

ij

j Sij

j S

t VQ t

V Q

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

1 1 1 11 2 3, 1 , , , 11

* * , , , 11 1 , 1/ 2 , 1/ 2

3

,1

ˆ ''

' ( ) , ( 1, , )

m m m mm mi k i k i k i km n ni i

i k k k j j j i i k i i k i kk j i k i k

m mj i m

h j j i i k b zj ij

T T T TT TV u S T u S T V Q A t

t z z

T Tt S V Q k k N

(Modified Courant number condition)

Mass conservation• Budget – vertical and horizontal sums• movement of free surface: violation of volume conservation

29Solar radiation

1 2/ /

0

1ˆ , ( ) (0) Re (1 R)eD d D d

p

HQ H z H

C z

0

0

(0)ˆp

HQdz

C

total solar radiation

Budget

The source term in the upwind scheme (F.D.)

, , 1 , , 1

0 , 0

,

( )ˆ

0b

m m m mi k i k i i k i km i

i ip i k p

mi k

H H A H HVV Q

C z C

H

to ensure total heat is conserved (black body)

This can cause overheating in shallow depths: adjust albedo

30Transport: TVD (1) 1

*

'm mi i j jn

j Vi

tT T Q T

V

Start from discretized advection

Now include downwind component * * *ˆ ˆ

2j

j j jD jT T T T

Flux limiter function (depends only on face j);Downwind and central schemes;The key is to find an appropriate function that doesnot violate max principle

0 2j

upwind

1*

' ' ˆ| | ( ) ( )2

' ' | | 1 ( ) | | ( )

2 2

jm mi i j j i j jD j

j S j Si i

j jmi j j i j i j

j S j Si i

t tT T Q T T Q T T

V V

t tT Q T T Q T T

V V

upwind

Sum of coefficients = 1; so max principle is satisfied if and only if the coefficient of Ti is non-negative

Put the last term in similar form as the upwind term

| | ( )' ' 2| | ( )2

| | ( )

jj i j

jj i j

j S j Si i

j j ij S

Q T Tt tQ T T

V V

Q T T

' '| | ( ) | | ( )

2j

j i j j j ij S j Si i

t tQ T T Q T T

V V

31

| |1 2| |2 2| |

| |

| |

jj i j

j jj i j

j S j S j S jm m i

m S

m m im S

j

j i j

Q T TQ T T

rQ T T

Q T Tr

Q T T

Transport: TVD (2)

1 ' '| | 1 ( ) | | ( )

2 2

' ' 1 | | 1 | | 1

2 2

j jmi i j j i j j i

j S j Si i

j ji j j j

j S j Si i

t tT T Q T T Q T T

V V

t tT Q Q T

V V

Max principle is satisfied if (Courant number condition)1 0

2

'1 | | 1 0

2

j

jj

j Si

tQ

V

Upwind ratio rm

ij

S+ S

Since 0 2, 0 2, 0jj

j Sj

Nr

The Courant number condition is '(1 ) | |

i

jj S

Vt

N Q

'| |

i

jj S

Vt

Q

as compared to upwind

32Transport: 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, ))

max(0,min(1, ))

( ) max(0,min(2, ))

| |

1 | |

r r

r

r r

r r

r

Super Bee (compressive)Minmod (diffusive)Osher

Van Leer

Second order TVD region(Sweby 1984)

33Turbulence closure (1) 2 2

k

Dk kM N

Dt z z

2 21 3 2 wall

Dc M c N c F

Dt z z k

0

0

0, or

,

,

k

kz h

z

n z hz l

n zz l

0 2

0

00

1, or

( )

, or

( ) ( ) , or p m n

k z hc z

l z h

c k z h

u

and

Essential b.c.

Natural b.c.FE formulation• Use natural b.c. first and essential b.c. is used to overwrite the solution at boundary• Neglect advection• The production and buoyancy terms are treated explicitly or implicitly depending on the sum to enhance stability• k and are defined at nodes and whole levels, and diffusivities at half levels

1 11 11 1

1 1 1/ 21

1 11 1 1

1 1 1/ 2

2 21

1/ 2

2 21

1/1/ 2

( ) 2 2 ( )6

( ) 2 2 ( )6

2

6

n nn n n nl l l

z l l l l k ll

n nn n n nl l l

b l l l l k ll

nl

l

zl

n ll

z k kl N k k k k t

z

z k kl k k k k k t

z

zM N

l N tz

M Nk

A

A

A( )

0 3 1/ 2 1 1 1111/ 2

1 112

2 2

1/ 20 3 1/ 2 1 1

1/ 22 2 1 1

11/ 21/ 2

( ) (2 )6(2 )

2( ) ( ) (2

6(2 )6

n n nll lln n n

l l

nl

l n nlb l lln n nl

l ln ll

zc k l k k

k k

zM N

zl k t c k l k k

zM N k k

k

A 11 )n

( , , )b zl k N

34Turbulence closure (2)

1 11 11 1

1 1 1/ 21

1 11 1 1

1 1 1/ 2

2 211 3 1/ 2

1/ 2

1

( ) 2 2 ( )6

( ) 2 2 ( )6

2

6

n nn n n nl l l

z l l l l ll

n nn n n nl l l

b l l l l ll

nn

l

ll

lz

zl N t

z

zl k t

z

zc M c N

k

zl N t

A

A

A( )

2 2 1 11 3 11/ 2

1/ 2

0 3 1/ 2 1 1 112 11/ 2

2 21 3 1/ 2

1/ 2

2 21 3 1/ 2

1/ 2

(2 )

( ) (2 )6

2

( ) (6

n n nl ln l

l

n n nlwall l ll

nn

l

ll

nl

b n ll

c M c Nk

zc c k l F

zc M c N

k

zl k t c M c Nk

A

1 11

0 3 1/ 2 1 1 12 11/ 2

2 )

( ) (2 )6

n nl l

n n nlwall l ll

zc c k l F

( , , )b zl k N

35GOTM 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?

www.gotm.net

36Numerical 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 diffusion number condition Upwind and TVD transport – Courant number condition Coriolis – stable but prone to “modes”

20.5

t

x

'1

g h t

x

1 2

3

1 1 11 2 32 0n n nu u u

37Lateral b.c. and nudging

Variable Type 1 (*.th)Time history; uniform across bnd

Type 2 (param.in);Const.

Type 3 (param.in);

Type 4 (*3D.th);Time history

Type -1 (param.in);radiation

Type -4 (*3D.th);nudging

Nudging near bnd

tides

S&T Clamp at i.c.

Nudge to i.c.

Nudge to *3D.th

u,v Via discharge

Via discharge

Tides (uniform across bnd)

Flather Nudge to uv3D.th (2 separate relaxation for in & outflow)

38Summary

ELCIRC SELFE

Time stepping Semi-implicit Semi-implicit

Horizontal grid Orthogonal unstructured unstructured

Vertical grid Z-coordinates Hybrid SZ-coordinates

Numerical algorithm Finite difference / finite volume

Finite element / finite volume

Convergence rate for non- orthogonal grid

Divergence At least 1st

Advection (momentum) ELM ELM (optional higher order Kriging)

Advection (transport) ELM/upwind ELM/upwind/TVD

Volume conservation Enforced Not enforced (but good)

Wetting/drying Yes Yes