Boundary conditions are then enforced on the boundary point B, computed by solving (via a bisection method) the 1D nonlinear equation (phi(B =0) on the segment between points G and (G_{1} = G-sqrt{2}, h, boldsymbol {n}_{G}), with h being the spatial step.
In details, we compute the outward unit normal vector (boldsymbol {n}_{G}) by discretizing Eq. (16) on G using a standard finite-difference scheme; then we apply the bisection method to solve the 1D nonlinear equation (phi(B =0) on the segment between points G and (G_{1} = G-sqrt{2}, h, boldsymbol {n}_{G}), with h being the spatial step.
In (39), E is the electric field, H is the magnetic field strength, σ, μ, and ϵ are the conductivity, the permeability, and the permittivity, respectively, of the domain, k is the time step, and h is the spatial step in all directions.
Let (t_{i}), (i = 1,2,ldots) ; (x_{n}), (n = 1,2,ldots) be the grids of the integral domain, (Delta t = t_{i + 1} - t_{i}) be the time step length, and (Delta x = x_{n + 1} - x_{n}) be the spatial step length.
We remark that the physical side length of the simulation domain is given by λ D · dx · N where (21) λ D = D A D P k d = D A T P δ · k d denotes the diffusion length of the activator, dx is the spatial step size of the discretization and N = 176 is the number of grid points which we kept fixed for all simulations.
The size of the FDTD scenario for this example is of 256 × 256 cells with the source located at the coordinates and being the spatial and time steps and respectively.
Let h and τ be the spatial and temporal step sizes, respectively.
This method is very fast and accurate even when the spatial step size h is large (h⩾0.1).
Suppose M is a positive integer, choose the spatial step h = 2a/M, and define x j = −a + j h, j = 0, 1,..., M. Let F j k be the approximation of Vdip(x j ) * |ϕ(x j, t k )|2, and ϕ j k be the approximations of (x j, t k ), which are the exact solution of (15 - 18) at the mesh grid (x j, t k ).
Typical parameter values for the spatial step were Δ x=Δ y=5 μm, and Δ t=0.01 s.
It can be seen from Table 4 that the convergence order of the compact difference scheme (2.1 - 2.3 2.1 - 2.3 4 wish respect to the spabout step size.
