Step 2: Operator Splitting Method For Sub-Layer Models

This is the section that we actually solve the two layer channel model, (5.46), (5.47) to update each layer solution. This involves two steps which we call stages. In stage one, we solve each model without the exchange terms. In this stage, the upper layer takes into account the flow exchange between the channel and the floodplains. This is easily achieved through the computation of lateral fluxes in the 2D channel cells adjacent to the floodplain cells, see figures 5.6 and 5.7(b). The second stage is where we resolve the flow exchange between the lower and upper layer flows in the channel. We adopt the operator splitting method for this two-stage process.

Step 2, stage 1: Sub-layers’ intermediate solutions and channel/floodplain interaction

As mentioned above, this stage solves the channel flow model without the exchange terms and at the same time, resolves the flow exchange between the channel and the floodplains through the lateral fluxes in the upper layer channel model. To proceed, we adopt the operator splitting approach as follows: Let (A1,i, Q1,i)n+1∗ be

the approximate solution of the 1D lower layer model, (5.46) without the exchange terms, namely

∂tA1+∂xQ1 = 0,

∂tQ1+∂xQ12/A1 =−gA1∂xη,¯

(5.68)

with the given initial data, (A1,i, Q1,i)n. Also, let (h2, ~q2)n+1_i,j ∗ be the approximate

solution of the upper layer 2D models,(5.47), without the exchange terms, namely ∂th2+∇.~q2= 0,

∂t~q2+∇.Fq(h2, ~q2) =−gh2∇(zb+h1),

(5.69)

given the initial data (h2, ~q2)ni,j. Fq(h2, ~q2) = (F2x, F y 2), withF2x = ( q2 2x h2+ g 2h 2 2, q2xq2y h2 ) T_, F₂y = (q2xq2y h2 , q2 2y h2 + g

2h22,)T. Recall that solving the above upper layer system, (5.69)

involves computing the fluxes across all cell faces, in particular, the lateral fluxes between the channel and the floodplains are computed in obtaining (h2, ~q2)n+1_i,j ∗ for

all channel 2D cells,Tij adjacent to the floodplains. This way, the flow interaction

between the channel and the floodplains are resolved.

In the remainder of this chapter, we will refer to the solutions of the sub- models without the exchange terms, (5.68) and (5.69), namely (A1,i, Q1,i)n+1∗ and

they (the intermediate solutions) have been obtained using an appropriate PDE solver. We postpone their solution methods to sections 5.6.6-5.6.8.

Step2, stage 2: Sub-layers’ Exchange and Update

In this step (stage 2), we resolve the exchange between the two layers and up- date each layer. Suppose we have the intermediate solutions, (A1i, Q1i)n+1∗ and

(h2, ~q2)n+1_i,j ∗, then the approximate solution of the 1D lower layer models including

the exchange terms, (5.46) is the approximate solution of the system ∂tA1,i=− Z Sdy, ∂tQ1,i=− Z uη1Sdy, (5.70)

given the initial data, (A1,i, Q1,i)n+1∗, and the approximate solution of the 2D mod-

els, (5.47) including the exchange terms, is the approximate solution of the system ∂th2,i,j =S,

∂t~q2,i,j =~uη1S,

(5.71)

given the initial data, (h2, ~q2)n+1_i,j ∗. Where~uη1 = (u(X~ , t), v(X~ , t))

T z=η1 . With the intermediate solutions, (A1i, Q1i)n+1∗ and (h2, ~q2)n+1_i,j ∗, then using forward Euler

time discretization, we approximate the solution of (5.70) and (5.71) as An+1_1,i =An+1_1,i ∗−S_iA∆t,

Qn+1_1,i =Qn+1_1,i ∗−S_iQ∆t, (5.72)

and

hn+1_2,i,j =hn+1_2,i,j∗+Si,j∆t,

~

qn+1_2,i,j =~q_2,i,jn+1∗+~uη1,i,jSi,j∆t,

(5.73)

respectively, where

Si,j ≈S|Ti,j, u~η1,i,j ≈~uη1

Tij , S_iA≈ Z Ki Sdy and SQ_i ≈ Z Ki uη1Sdy.

To complete the description of the scheme, we need to define the exchange terms and interface velocities;Si,j, ~uη1,i,j, S

A i andS

Q

terms. However they can be determined by respecting the following conditions : (i) Conservation of mass and momentum.

(ii) Discrete version of (5.32).

(iii) Discrete consistency requirement (definition 5.6.1) of the resulting solution and (iv) And non-negativity of water heights.

The above conditions allow to first obtain the heights, hence the exchange terms and interface velocities are calculated.

Lower layer Area

For global conservation, the total mass/momentum leaving one layer must equal the mass/momentum entering the other layer. Hence we assume the following :

S_iA=X j Si,j∆yij and SiQ= X j uη1,i,jSi,j∆yij. (5.74) Recall An+1_i :=An+1_1,i +X j hn+1_2,i,j∆yj. (5.75)

Substituting the expressions for An+1_1,i and hn+1_2,i,j in (5.72) and (5.73) into (5.75), making use of the first conservation property, (5.74), then (5.75) gives

An+1_i =An+1_1,i ∗+X

j

hn+1_2,i,j∗∆yij. (5.76)

This explicitly gives the total wetted area, An+1_i directly from the intermediate solutions (n+1* values). Next, the discrete version of (5.32) is

An+1_1,i = min

An+1_i , Ac,i

. (5.77)

Upper layer Heights

To compute the upper layer heights, we use the definition ofAn+1_1,i in (5.77) along- side the relations (5.75) and (5.76). Before we continue, lets define the following notations, An+1_2,i ∗ := P

jh n+1∗

2,i,j ∆yij and An+12,i :=

P

jh n+1

2,i,j∆yij. We consider the

Case 1 : If An+1_1,i =An+1_i (Lower Layer at tn+1 not full) Then by (5.75), we haveP jh n+1 2,i,j∆yij =An+1i −A n+1 i,i = 0. Sinceh n+1 2,i,j ≥0, hence hn+1_2,i,j = 0. (5.78)

Case 2 : If An+1_1,i =Ac,i (Lower Layer attn+1 full)

Then, subtracting (5.76) from (5.75) gives

An+1_2,i =An+1_2,i ∗+

An+1_1,i ∗−Ac,i

(5.79) and consider two further cases.

Case 2a : IfAn+1_1,i ∗−Ac,i ≥0 (Lower Layer full at intermediate state,tn+1∗)

Then An+1_2,i ≥ An+1_2,i ∗ by an amount A_excess,in+1∗ := An+1_1,i ∗−Ac,i ≥ 0, so we add the

constant excess height, hn+1_excess,i∗ = A_excess,in+1∗ /Bi to the intermediate solution hn+1_2,i,j∗,

namely hn+1_2,i,j=hn+1_2,i,j∗+A n excess,i Bi . (5.80)

Case 2b : If An+1_1,i ∗−Ac,i <0 (Lower Layer not full at intermediate state,

tn+1∗)

Then An+1_2,i < An+1_2,i ∗ by the amount, A_gap,in+1∗ =Ac,i−An+11,i ∗ <0. Hence we remove

An+1_gap,i∗ from An+1_2,i ∗ using the following algorithm. • initializehn+1_2,i,j =hn+1_2,i,j∗ for allTij.

• hn+1_gap,i∗ = A n+1∗ gap,i Bi , T OL= 10 −12_. • while(hn+1_gap,i∗ > T OL) i for allTij (a) ht=hn+1_2,i,j.

(b) Reduce upper layer height by the gap height :

hn+1_2,i,j =max(0, h_2,i,jn+1−hn+1_gap,i∗). (5.81)

(c) Remove area of reduced height from total gap area: An+1_gap,i∗ =An+1_gap,i∗− |hn+1_2,i,j−ht|∆yi,j.

ii hn+1_gap,i∗ = A

n+1∗

gap,i

Bi .

Equations (5.78), (5.80) and (5.81) compute hn+1_2,i,j in Tij ∈ Ki for all three cases

above.

Computing the Exchange Terms and the Interface velocity

Having computed hn+1_2,i,j , we now compute the exchange terms, Si,j using the first

equation in (5.73), hence

Si,j =

hn+1_2,i,j−hn+1_2,i,j∗

∆t . (5.82)

Next, we compute the interface velocity,~uη1,i,j. As proposed for multilayer systems in [Audusse et al., 2011], if water moves from lower layer to upper layer (Si,j >0),

then the interface velocity,~uη1,i,j is that of the lower layer. But if the reverse is the case (Si,j ≤0), then ~uη1,i,j is that of the upper layer. Hence we define

~ uη1,i,j:=        ~un+1_2,i,j∗= ~q n+1∗ 2y,i,j hn_2,i,j+1∗, ifSi,j≤0 (un+1_1,i ∗,0)T = Qn_1,i+1∗ An_1,i+1∗,0 T , ifSi,j>0. (5.83)

Computing the Discharges

With the above definitions for the exchange terms and the interface velocity, then the upper layer discharge,~qn+1_2,i,j is computed in (5.73), namely

~

q_2,i,jn+1 =~qn+1_2,i,j∗+~uη1,i,jSi,j∆t. (5.84) As stated above, to conserve momentum the lower layer exchange term,S_iQis defined as S_iQ= N1 y X j

uη1,i,jSi,j∆yij, see (5.74). (5.85)

Hence, the lower layer discharge is computed as

Qn+1_1,i =Qn+1_1,i ∗−S_iQ∆t, see (5.72). (5.86)

This ensures that momentum is conserved. Hence, the discharges for both layers are obtained. Therefore, we now have (A1,i, Q1,i)n+1, and (h2, ~q2)n+1i,j . With these

results, we update the full 2D data, (Hi,j, ~qi,j)n+1 in the next section which is the

third and last step.