Initial and boundary conditions Stability condition
explicit determination of U
i
is written as U
p i
¼ U
n i
¹ Dt
Dx F
n i þ 1=2
¹ F
n i ¹ 1=2
þ DtS
n i
8 where F
iþ12
n represents the numerical flux through the cell face between nodes i þ 1 and i. Dx is the grid spacing and
Dt is the computational time step. All the terms on the right-hand side of eqn 8 are evaluated at the known
time level n and, therefore, U
i
can be computed explicitly. The numerical flux F
iþ12
is computed using the following formula.
F
i þ 1
2
¼ 1
2 [
F
R
þ F
L
¹ a
U
R
¹ U
L
] 9
in which, a ¼ a positive coefficient, F
R
¼ fU
R
the flux computed using the information from the right side of the
cell face and F
L
¼ fU
L
¼ the flux computed using the information from the left side of the cell face. U
L
and U
R
are obtained using a MUSCL monotone upwind scheme for conservation laws approach.
U
L
¼ U
i
þ dU
i
2 10
U
R
¼ U
i þ 1
¹ dU
i þ 1
2 11
There are several ways of determining dU
i
and dU
iþ1
using different
slope limiter
procedures
6,35
. The
‘minmod’ limiter is followed in the present study. Accord- ing to this,
dU
i
¼ minmod
U
i þ 1
¹ U
i
, U
i
¹ U
i ¹ 1
12 where the minmod function is defined as
minmod a
, b
¼ a
if lal , lbl and ab . 0
b if
lbl , lal and ab . 0 if ab 0
8 :
13 The positive coefficient a is determined using the maxi-
mum value for all the grid points of the largest eigen value of the Jacobian of the system of equations. This is
given as
a max q
i
h
i
þ
gh
i
p i ¼ 1 to N
14 where, N is the total number of grid points. The vector
equation eqn 8 gives the predicted values of h and q at the unknown time level at any node i.
3.1.2 Corrector part The vector U at the unknown time level n þ 1 and at node i,
i.e. U
i nþ1
is computed using the predicted values and the values at the time level n.
U
n þ 1 i
¼ 0:5 U
n i
þ U
p i
¹ Dt
Dx F
p i þ 1=2
¹ F
p i ¹ 1=2
ÿ þ DtS
p i
15 where
F
p i þ 1=2
¼ 1
2 [
F
p R
þ F
p L
¹ a
U
p R
¹ U
p L
] 16
Following the recommendations of Alcrudo et al.
6
, U
R
and U
L
are determined from U
iþ1
and U
i
using the same dU
i þ 1
and dU
i
as determined in the predictor step. This procedure results in better numerical stability.
In eqn 15, only the source term in the momentum equa- tion the second component of the vector equation is eval-
uated using the predicted values of h and q. However, the source term in the continuity equation i.e. R ¹ I
i
is eval- uated using the values at the known time level instead of the
predicted values. Strictly speaking, this procedure decou- ples the subsurface flow computations and the surface
flow computations during the computational time step Dt. However, the response of the subsurface flow to the varia-
tion in surface flow depth is much slower than the response of surface flow to changes in the rate of infiltration
5
. There- fore, the above decoupling does not affect the results sig-
nificantly. In fact, numerical experimentation showed that determination of the infiltration rate, I, during the corrector
step by using the predicted flow depth did not alter the results. On the other hand, the decoupling procedure
resulted in significant savings of the computational time, since the subsurface flow is computed only once during a
time step.