93
matrix A. Discretizing this form in time as well yields:
‰ι = ⅛ + ɪ(z - ɪʌfi - ɪɪf')df
C√ Jλ,c ιtm
(5.12)
where I is the identity matrix. This integration scheme is called ’’forward euler”, and is
stable as long as 2dt < τ. It is an explicit scheme that relies on the slope of the voltage at
the current time point t to predict the future voltage at t + 1. A scheme which gives better
stability is:
г / ɪ ι ∖r1
⅛= 1 r~Fa^ 7Γ^F1)
di + ⅛-ι) (5.13)
This scheme is called ’’backwards euler,” and an ’’implicit” scheme in that a system of
equations must be solved to find the voltage at time t from time t - 1.
This scheme can be expanded to model a two-dimensional network by changing A from
the tridiagonal second difference form to the difference matrix for the two dimensional
laplacian operator v2. This matrix can be computed by taking the sum of the Kronecker
tensor products of the identity matrix and one-dimensional second difference matrix plus
the tensor product of the one-dimensional second difference matrix with the identity matrix
[81].
Finally, the steady-state distribution can be solved by eliminating the capacitance from
this model, where A can represent the Id or 2d difference matrix:
1 1 ∖ ʌ ^
— I^ι AI vss — i (5.14)
ʃɪm ʃɛe /
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