Indian Institute of Science / A large number of mathematical models in engineering and physical sciences employ Partial Differential Equations (PDEs). The sheer number of operations required in numerically integrating PDEs in these applications has motivated the search for faster methods of computing. The conventional uniprocessor computers are often unable to fulfill the performance requirements for these computation intensive problems. In this dissertation, a cost-effective message-based multiprocessor system which we call the Broadcast Cube System (BCS) has been proposed for solving important computation intensive problems such as, systems of linear algebraic equations and PDEs. A simulator for performance evaluation of parallel algorithms to be executed on the BCS has been implemented. A strategy (task assignment . algorithm) for assigning program tasks with precedence and communication constraints to the Processing Elements (PEs) in the BCS has been developed and its effectiveness demonstrated. This task assignment algorithm has been shown to produce optimal assignments for PDE problems. Optimal partitioning of the problems, solving systems of linear algebraic equations and PDEs, into tasks and their assignment to the PEs in the BCS have been given. Efficient parallel algorithms for solving these problems on the BCS have been designed. The performance of the parallel algorithms has been evaluated by both analytical and simulation methods. The results indicate that the BCS is highly effective in solving systems of linear algebraic equations and PDEs. The performance of these algorithms on the BCS has also been compared with that of their implementations on popular hypercube machines. The results show that the performance of the BCS is better than that of the hypercubes for linear algebraic equations and compares very well for PDEs, with a modest number of PEs despite the constant PE connectivity of three in the BCS. Finally, the effectiveness of the BCS in solving non-linear PDEs occurring in two important practical problems, (i) heat transfer and fluid flow simulation and (ii) global weather modeling, has been demonstrated.