Introduction

In the first part of this work, a description of the mathematical and numerical models employed is presented.

The mathematical model implemented in numerical form solves the 2D hydrodynamic equations for shallow waters, obtained by vertically integrating pressure-velocity 3D Navier Stokes equations.

A finite differences scheme is being employed over a staggered mesh to solve the discrete form of the shallow water equations. As a result of this scheme, a tri-diagonal system must be solved. A block domain decomposition technique with overlapped regions is used to parallelize the program.

Two approaches are employed for the parallel solver:

• a fully implicit method: every process, working on a block, builds its part of the tridiagonal matrix and a distributed tri-diagonal (Johnsson et al., 1985) implementation solves the system. Data on the overlapped regions are sent forth and back between adjacent blocks in each time step.
• an explicit-implicit method (or block-wise implicit): every process builds its own tridiagonal system using an explicit iteration to predict a boundary condition on the overlapped cells.

The advantages of each method are studied and explained below.

A stability analysis of the the scheme is performed in one and two dimensions employing Fourier transform of the constant coefficients shallow water PDE. Condition for both time and space step (cell size) in the discrete solution are found.

In the second part of this work, extensive applications of the above mentioned models are made to test speedup, accuracy and convergence of the numerical results to both analytical solutions and real world data.

A comparison of the numerical results and the analytical solution, for the one dimensional case, is carried out. Comparison of results from the extensive application of the code to a real tide simulation over the Río de la Plata are presented in chapter 8 and a graphical output of an application to predict the behavior of oil spills is introduced.

The speedup of a parallel program, defined as the serial runtime (ts) over parallel runtime (tp), is measured using several grids resolutions on the same geometry: a rectangular channel. The meshes employed in this study have 5.000, 20.000 and 80.000 grid points.

Numerical accuracy validation is performed using a grid refinement study over the same meshes.

The "parallel computers" available to test the performance have been:

1.

A Bull DPX/20 Escala SMP (symmetric multi processing) using the shared memory for message passing (4 Power PC 601).

2.

Same of above but employing sockets message passing,

3.

A "parallel virtual machine", formed by a cluster of RISC workstations running UNIX and connected through a FDDI LAN

4.

A "parallel virtual machine", built by a cluster of PC's running UNIX and passing messages through an unswitched Ethernet LAN.

The RISC workstations cluster have shown advantages over the SMP equipment, and other parallel computers, in the performance/price ratio, scalability, standard components usage, as it's documented in Warren et al. (1997). That has been confirmed along this work. The advantages are more evident in the case of the cluster of PCs thanks to the components low price.