## Ground-water Hydrologic Model (GHM)

The ground-water domain in a watershed is described in above figure. The continuous domain is discretized into a set of small rectangular cells with dimensions z, x, and y. z is the thickness of aquifer (vertical), and x and y are cell dimensions in the x and y coordinate directions. Although the flux can be input to the ground-water domain in GHM, non-flow boundary of the domain is assumed to be corresponded with watershed boundary derived from available DEM or digitized from maps. Non-flow boundary is also assumed at a certain depth in the vertical direction. This distributed depth in each cell is handled in HMS with a similar way as DEMs. This vertical boundary should be the boundary between the local ground-water flow system of the simulated watershed and regional ground-water flow system.
The finite-difference numerical method is used to provide a solution to above equation. After the discritization of ground-water domain, sets of tridiagonal matrices can be assembled for x and y direction or for row and column. The iterative alternating direction implicit method (Peaceman and Rachford, 1955; Prickett and Lonnquist, 1971; Yu and Schwartz, 1995) is applied to solve the matrix system. At any given time step, the method reduces a large set of simultaneous equations down to a number of small sets. The node equations of an individual column of the domain are solved while all adjacent related terms are kept constant. This means that the set of column equation is implicit in column direction and explicit in row direction. Two tridiagonal matrix solvers, forward solution and backsubstitution (FB), and reduction and backsubstitution (RB) (Yu and Schwartz, 1995; Yu, 1997), were implemented in GHM. The traditional FB method, is intrinsically data dependent, in the equations at a given grid node depend on values from adjacent grid nodes. This feature effectively prohibits vectorizing and parallelizing the method for solution on a vector and parallel processor. An alternative method of solution, that of reduction and backsubstitution (RB), has no data dependence and was implemented in Fortran for vector and parallel processors to solve a tridiagonal matrix system in GHM. The vectorized code has an overall execution rate of 110 MFLOPs on a Cray Y-MP that is six to 16 times faster than the scalar code. Implementing the code in vector-parallel mode using eight undedicated processors on a Cray Y-MP results in an overall additional speedup of 2.8 times in wall clock time for two-dimensional flow problem and 2.2 times for three-dimensional flow problem (Yu, 1997). In comparison with FB code, RB code requires 26% less of CPU time and 66% less of wall clock time on same flow problem. The implementation of this module isdescribed in detail by Yu and Schwartz (1995), and Yu (1997).