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 approximation to a second-order derivative may be obtained using only three points in 2-D (the vertices of a triangle), four points in 3-D (the vertices of a tetrahedron), and in general, N + 1 points in a space of dimension N. A grid using N + 1 points to compute derivatives is called minimal. The use of minimal grids does not introduce any complication in programming and suppresses some artifacts of the nonminimal grids. Some numerical tests in 2-D show that the propagation of waves is as accurate as when performed with conventional grids. -from Authors