Signal coupling in nonlinear hybrid optical structures: A numerical approach

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

“…However, it is clear from Figure 9a that the standard MoC-ME, whose stencil is given by the three circles in Figure 1, cannot be used for this purpose. The key trick that enables the use of the MoC-ME is to employ the rotated stencil, as shown in Figure 9b and c. Indeed, the solutions (y ± ) 1 1 and (y ± ) 3 1 have already been found at the previous steps. Then, rotating the stencil shown in Figure 9b by −90 • , one obtains the standard MoC-ME stencil in Figure 9c.…”

“…It will contain the trick with the "rotated stencil" as in the MoC-pRK3 case, as well as an additional twist. To find ( y + ) 5 2 , one requires (y ± ) 3 0 . In analogy with the MoC-pRK3 case, it needs to be computed with error O(h 4 ).…”

“…Similarly, if one instead uses the rotated stencil, that would require the knowledge of ( y + ) 4 0 , which is not available, either. Therefore, the only available option is to find (y ± ) 3 0 is by the MoC-pRK3 with the rotated stencil, as shown in Figure 11b and c. In this stencil, we already know the solution at nodes (n, m) = (1, 2), (2, 1), (3,2), and (4, 1). However, we do not know the solution y + at node (5,2), because this is precisely the solution that we need (y ± ) 3 0 for!…”

“…[4,5,[20][21][22][23][24][25][26][27][28][29]44-64] there); see also Refs. [2][3][4][5][6][7][8][9][10][11][12][13] here and Ref. [14] about applications to power transmission lines.…”

“…Therefore, the accuracy of an MoC-based scheme follows that of the ODE solver. Our goal is to develop schemes of order higher than two for the quasi-ODE system (3). We emphasize the word "quasi": the fact that each of the ODEs in (3b) is solved along a different characteristic introduces several nontrivial modifications to the standard procedure of solving a system of (true) ODEs numerically.…”