Section: New Results

Discretization of the Laplacian operator using a multitude of overlapping cartesian grids

A new finite-difference approach to solve a Laplacian operator has been developed, using patches of overlapping grids where a fined level is needed, leaving coarser meshes in the rest of the computational domain. These overlapping grids will have generic quadrilateral shapes (as shown in figure 11).

Figure 11. results

(a) Exact: $sin({(x-0.5)}^2+{(y-0.5)}^2)$ (b) Evaluated (c) Numerical grid (d) Numerical error

A monolithic approach is used to solve the algebraic equations, applying restriction and prolongation operators to fill the non-diagonal blocks of the resulting matrix. These operators works on data structures communicated between the different grids using ad hoc parallel inter-communicators, as shown in figure 12.  12.

Figure 12. Monolithic approach

Works are going on to change the solver from the finite-difference approach to a finite-volume one, and to implement the 3D case. The use of a finite-volume solver can benefit from the usage of octree patches instead of cartesian grids, obtaining a more accurate refining and a greater precision.