The finite-difference framework defines generic operators for the first and second derivative along a given direction, and we declare the structure named TripleBandLinearOp, which contains the lower, diag and upper arrays with the correct values (for instance, the coefficients of \(f(x_{i-1})\), \(f(x_{i})\) and \(f(x_{i+1})\) in the equation for \(\frac{\partial f}{\partial x}(x_{i})\)) based on the 2-D mesher. Here, we also define a structure named NinePointLinearOp which contains the nine neighbors, managing the stencils arguments used in the cross-defivative \(\frac{\partial^{2} f}{\partial x_{i} \partial y_{j}}\) along two directions. The TripleBandLinearOp and NinePointLinearOp are shown in the following figure. a represents x direction, b represents y direction, and c represents nine neighbors.
You might find the details about differential operator and evolution scheme introduced in Internal Design of Finite-difference Hull-White Bermudan Swaption Pricing Engine.