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.

3.2 Initial and boundary conditions

Values of the flow depth, the discharge and the infiltration rate are specified at all the nodes at time t ¼ 0 as the initial conditions. The initial infiltration rate is equal to the rainfall rate. Although the initial flow depth and the discharge are equal to zero overland flow on an initially dry surface, a very thin water film of depth h ini and corresponding uniform flow discharge, q ini are assumed to exist at time t ¼ 0. This assumption is made to overcome the numerical singularity in a simple way. However, it should be noted that the out- flow hydrograph may be sensitive to the h ini value and, therefore, it should be chosen as small as possible. The explicit finite-difference scheme described earlier can be used to determine h and q at the unknown time level only at the nodes i ¼ 2 to N ¹ 1. The values of the variables at the upstream and the downstream ends of the domain are determined using the appropriate boundary con- ditions. The discharge at the upstream end is equal to zero. However, the discharge at the upstream end is specified as q ini to be consistent with the initial conditions. The flow depth at the upstream end can then be determined using the negative characteristic equation 11 . In the present study, a simple extrapolation procedure is adopted to deter- mine the upstream flow depth from the depth at the interior nodes. Numerical experimentation in the initial stages of the model development showed that the extrapolation pro- cedure gave satisfactory results. Similar extrapolation pro- cedure is adopted to determine h and q at the last node N. 570 V. Singh, S. M. Bhallamudi

3.3 Stability condition

The high-resolution Lax–Friedrichs scheme adopted in the present study is an explicit scheme. Therefore, the computa- tional time step, Dt, is chosen using the CFL condition. C n ¼ Dt Dx g h þ  gh p h i 1:0 17 in which C n ¼ Courant number. Dt is chosen dynamically in the numerical model such that eqn 17 is satisfied at all the nodes i ¼ 1,2 … N.

3.4 Subsurface flow