Henry Juang 1

Non-iteration Semi-Lagrangian:mass conservation

Hann-Ming Henry JuangEnvironmental Modeling Center, NCEP

NOAA/NWS

April 18, 2007

Henry Juang 2

Introduction

• The new semi-Lagrangian scheme– Without iteration to find departure/mid points– Combine two 1-D interpolation/remapping– Without halo in global model– Encouraging results from regional/global tests

• The concerns– Can we have conservation?– Positive advection?

Henry Juang 3

€

∂ρ∂t

+∂ρu

∂x+

∂ρv

∂y= 0

∂ρ

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟X −direction

+∂ρu

∂x+

∂ρ

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟Y −direction

+∂ρv

∂y= 0

For mass conservation, let’s start from continuity equation

Consider 1-D and rewrite it in advection form, we have

€

∂ρ∂t

⎛

⎝ ⎜

⎞

⎠ ⎟X −direction

+ u∂ρ

∂x= −ρ

∂u

∂x

Advection form is for semi-Lagrangian, but it is not conserved if divergence is treated as force at mid-point,So divergence term should be treated with advection

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Divergence term in Lagrangian sense is the change of the volume if mass is conserved, so we can write divergence form as

€

∂u

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟Lagrangian _ sense

=1

Δx

dΔx

dt

€

dρΔx

dt

⎛

⎝ ⎜

⎞

⎠ ⎟X −direction

= 0

∂ρΔx

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟X −direction

+ u∂ρΔx

∂x= 0

Put it into the previous continuity equation, we have

which can be seen as

€

ρΔ x( )departure= ρΔx( )arrival

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AD M€

XD = XM −UM Δt

XA = XM + UM Δt

ρ A*n +1 = ρ D

n−1

Interpolation

remapping

€

ρDn−1

relocation

€

ρA*n +1

€

ρ*n +1

Instead doing following

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ALDL M

€

XLD = XL

M −ULM Δt

XRD = XR

M −URM Δt

ΔD = XRD − XL

D

Interpolation

€

ρDn−1

relocation

€

ρA*n +1

We do

X

DRAR

€

ρDn−1ΔD = ρ A

n +1ΔA

€

ΔD

€

ΔA

ML MR€

XLA = XL

M + ULM Δt

XRA = XR

M + URM Δt

ΔA = XRA − XL

A

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Since we are using cubic spline, the given value can be presented piece-wisely by

€

SDn−1(x)dx

DL

DR∫ = SAn +1(x)dx

AL

A R∫€

ρ =S(x)

so the previous mass equality can be replaced as following

Also we want to make sure that total mass is conserved as

€

SRn−1(x)dx∫ = SD

n−1(x)dx∫ = SAn +1(x)dx∫ = SR

n +1(x)dx∫

This implies that mass conservation should be used during interpolation from regular grid to departure grid and from arrival grid to regular grid.

where subscript R is regular grid D is departure grid A is arrival grid for

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Interpolation with global conservation

€

X iD

€

X i+1D€

ρ jn−1

€

ρ j +1n−1

€

ρ jn +1

€

ρ j +1n +1

X X XX

€

ρ j−1n−1

€

ρ j +1n−1

X X XX€

SDn−1(x)dx

X iD

X i+1D

∫ = SRn−1(x)dx

X iD

X j+1R

∫ + SRn−1(x)dx

X j+1R

X i+1D

∫

€

X i+1A

€

X i+2A

€

X iA

€

X i−1A

€

X i−1D

€

X i+2D

€

X i−2D

€

ρ jn +1

€

ρ jn +1

€

SRn +1(x)dx

X jR

X j+1R

∫ = SAn +1(x)dx

X jR

X iA

∫ + SAn +1(x)dx

X iA

X j+1R

∫

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Summary of the procedure

(1) Using

€

ρ jn−1

and cubic spline conditions, we can solve

€

SAn +1(x)dx

X iA

X i+1A

∫ = SDn−1(x)dx

X iD

X i+1D

∫

= SRn−1(x)dx

X iD

X j+1R

∫ + SRn−1(x)dx

X j+1R

X i+1D

∫€

SRn−1

( )j

(2) Using semi-Lagrangian mass conserved advection as following

and cubic spline conditions at arrival points, we can solve

€

SAn +1

( )i

(3) Using mass conservation interpolation as following

€

SRn +1(x)dx

X jR

X j+1R

∫ = SAn +1(x)dx

X jR

X iA

∫ + SAn +1(x)dx

X iA

X j+1R

∫ and cubic spline conditions at regular point, we can solve

(4) With we can have

€

SRn +1

( )j

€

ρ jn +1

€

SRn +1

( )j

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Cubic Spline Interpolation/Remapping

Definition of piece wise cubic spline curve fitting with periodic condition

which requires 4 equations to solve all 4 unknown coefficients.

€

S j (x j ) = ρ j <=== given

S j (x j +1) = S j +1(x j +1)

′ S j (x j +1) = ′ S j +1(x j +1)

′ ′ S j (x j +1) = ′ ′ S j +1(x j +1)

€

S j (x) = a j (x − x j )3 + b j (x − x j )

2 + c j (x − x j ) + d j

In case of interpolation, such as the 4 equations are

In case of remapping, such as

the 4 equations are the same as above, except replace the first one by

€

SRn−1

( )j

€

SAn +1

( )i or

€

SRn +1

( )j

€

S j (x)dxX j

X j+1∫ = A j <=== given where

€

j =1,2,3,......,N

Henry Juang 11

€

∂q

∂t+ u

∂q

∂x+ v

∂q

∂y= 0

∂ρ

∂t+

∂ρu

∂x+

∂ρv

∂y= 0

€

∂ρq

∂t+

∂ρqu

∂x+

∂ρqv

∂y= 0

∂ρq

∂t+ u

∂ρq

∂x+ v

∂ρq

∂y= −ρq

∂u

∂x+

∂v

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

dρqΔ

dt= 0

€

dρΔ

dt= 0

How about mass conservation for tracer ?

If we use tracer and continuity equation as following together

Then density weighted tracer can be treated as conservation as

Combine it with continuity equation, we can have conserved tracer advection

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Summary• Mass conservation can be obtained

– No guess, no iteration to find departure/mid points

– No halo is required but transpose, which exists in the code already

– Cheap1-D interpolation/remapping – Local and/or global conserved

• Test simple cases with success– tracer advection and steady rotational flow