A combined variational and multi-grid approach for fluid dynamics simulation

A new approach for solving Partial Differential Equations problems related to fluid dynamics problems is being proposed. The method merges two existing methods: Using the variational techniques of finding the Action Minimum and using the Multi-Grid techniques of solving the equation on a coarse grid, interpolating it to finer grids, performing reduced computer work on finest grid, receiving the same accuracy. The proposed method simplifies the Multi-level adaptive techniques (MLAT) ideas. The experiments has shown that solving the variation problem on a grid with n2 is much less expensive (more than factor 4) than that on a grid with (2n)² grid-points. The method solves a problem on a coarse grid, interpolates it to the finer grid and uses the interpolated values as initial values - better guess values for the problem on the finer grid. The finer grid has changed its role in the second step, to a coarse grid. This process improves the solution duration. The interpolation computer-work maybe performed efficiently once and therefore may be neglected compared to the time invested in the variation solution itself. In this work we improve on our previous results¹⁷ and also investigate further aspects of our method including the effects of introducing a better initial configuration ("guess")⁷, using the relaxation technique and the effect of the required accuracy on computation time. We also compare the resources needed for interior and boundary points by our algorithm.