Implicit Methods for Complex Cartesian Domains

The basic operator for the construction of these competent implicit finite-difference schemes has been proposed in [15]. Their staggered form, for f = x, y, z, is denoted as 

spatial derivatives are approximated to fourth-order through the matrix arrangement of 

FIGURE 2.3 Treatment of a general lattice for (a) five nodes or more and (b) four nodes 

For the time-marching procedure, the leapfrog integration may be substituted by the fourth-order Runge-Kutta one, which staggers the variables in space but not in time. Thus, for dU/dt = /( /), it is derived 

In general, boundary conditions are imposed at the end of each stage of (2.41) or the leapfrog time-step. Finally, in the case of absorbing boundary conditions, all derivatives are computed by the implicit algorithm across the entire domain including its interior and the absorber. Then, each system is updated by (2.41). 

---