FINITE-DIFFERENCES ON MINIMAL GRIDS

Conventional approximations to space derivatives by finite differences use orthogonal grids. To compute second-order space derivatives in a given direction, two points are used. Thus, 2N points are required in a space of dimension N; however, a centered finite-difference approxim ation to a second-order derivative may be obtained using only three po ints in 2-D (the vertices of a triangle), four points in 3-D (the vert ices of a tetrahedron), and in general, N + 1 points in a space of dim ension N. A grid using N + 1 points to compute derivatives is called m inimal. The use of minimal grids does not introduce any complication i n programming and suppresses some artifacts of the nonminimal grids. F or instance, the well-known decoupling between different subgrids for isotropic elastic media does not happen when using minimal grids becau se all the components of a given tensor (e.g., displacement or stress) are known at the same points. Some numerical tests in 2-D show that t he propagation of waves is as accurate as when performed with conventi onal grids. Although this method may have less intrinsic anisotropies than the conventional method, no attempt has yet been made to obtain a quantitative estimation.