In turn, we can approximate the nodal electric potential
φ(x,y,z,t) = | ∑ | N_{a}(x,y,z) φ^{a}(t) |
a |
and number of particles
n_{i}(x,y,z,t) = | ∑ | N_{a}(x,y,z) n_{i}^{a}(t) |
a |
at a time t_{n} by
and evaluating
u_{n} = | [ |
n_{+, n} n_{-, n} φ_{n} |
] |
in the Taylor series we obtain a discrete approximation in time
where β takes values from [0, 1] and Δt denotes time step. After incorporating it into a general form of time-dependent equations A_{{1, 2}}(u)
where K(u) represents these parts of A_{{1,2}}(u) with a space-dependent operator we get time approximation for a given node
K(N_{a} Ι u^{a}_{n}) + C N_{a} Ι | { | 1/(βΔt) (u^{a}_{n} - u^{a}_{n-1}) - (1 - β)/β du^{a}_{n-1}/dt | } | = 0. |
When β = 1 an approximate solution to the semi - discrete equations at each time t_{n} is given by the Euler ,,backward'' scheme
otherwise, the expression for u^{a}_{n} is as follows