| An integral method, conceptually simple but increasingly sophisticated in implementation, is used to solve a sequence of problems of increasing complexity, leading from the Laplace equation, through Poisson and Helmholtz, to the Navier-Stokes equations. The Direct, Formal Integration method involves three steps: (1) Formally integrate along a chosen trajectory. (2) Resulting Volterra-forms are studied for new physical, mathematical and numerical insights. (3) Solve the Volterra-forms numerically by an improved 'micro'-Picard iteration. The method appears equally facile for initial value, boundary value and asymptotic boundary value problems and Dirichlet and Neumann boundary conditions can both be incorporated with ease. Using DFI as an elliptic solver has the major advantage of total freedom in choice of 'sweep' patterns with only changes in 'DO-Loop' indices. Aside from simplicity, ease of implementation and its iterative nature, DFI is a global approach and is optimally compatible with computers due to minimal subtractions and divisions.;A large number of solutions was obtained for elliptic equations namely: Laplace, Poisson and Helmholtz. With these solutions to serve as a benchmark, the two dimensional incompressible Navier-Stokes equations were then solved for a range of Reynolds numbers, using primitive variables. For the calculation of pressure, three techniques were incorporated into the basic DFI scheme. Abdallah's method for calculating the pressure, with DFI, was computationally most efficient, for the channel flow problem. For the driven cavity experiment, DFI with 'SIMPLE' was the best. In all cases, the divergence-free constraint served as a check and as a convergence criteria.;Finite element and finite difference solutions for the two model problems were then compared with DFI results to evaluate accuracy and run times. The finite difference code used Abdallah's method for the calculation of pressure, and a simple explicit marching procedure for the momentum equations. The finite element program used parabolic, isoparametric quadrilateral elements and a frontal solution technique for assembly and solution of the resulting matrices. DFI has proved to be a valuable new tool for elliptics and possibly an optimal elliptic solver. |
