| A new approach to computational fluid dynamics is explored that attempts to solve the governing equations of fluid motion for solutions and a solution-adaptive grid simultaneously. The new approach is to define the residual (or fluctuation), a measure of the error in satisfying the governing equations, and minimize it with respect to both the solution values and the grid structure. It is discovered that the grid movement derived from this simple procedure faithfully responds to the physics of the governing equations: form a grid that is isotropic but responsive to singularities for equations of elliptic type and an anisotropic (characteristic or shock) grid for those of hyperbolic type. The mechanism of grid movement is extensively discussed for one-dimensional boundary value problems, and hyperbolic and elliptic partial differential equations in two dimensions. Computational results are presented for some simple but varied situations that are remarkably well-resolved on rather coarse grids. |
