Analog and Hybrid Computer Programming, 2nd Edition

B Solving the heat equation with a passive network

The two-dimensional heat equation as treated in section 6.30 can also be solved by means of a two-dimensional grid consisting of a number of identical passive resistor/capacitor (RC) nodes as shown below.²¹⁵ The aim is to model the heat flow in a two-dimensional sheet of thermally conductive material such as a thin metal plate. The flow is described by

$\dot{u}=\alpha {\nabla }^{2}u.(\text{B}.1)$

To model the behavior of this equation by means of a discrete two-dimensional grid of passive elements, it is approximated by finite differences yielding

${\dot{u}}_{i,j}=\alpha \left(u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-4u_{i,j}\right)+q_{i,j}$

with u_i,j denoting a single grid node, and q_i,j representing a heat source or sink ...