An electric circuit theory approach to finite difference stability

One of the major difficulties involved in the numerical solution of partial differential equations of the parabolic or hyperbolic type by means of finite difference approximations is the tendency of the solution to be unstable under certain conditions. If instability exists, a small error (e.g., round-off error) arising at some point in the computation procedure, tends to become larger and larger as the computation progresses, until the error terms completely overshadow the desired solution, making it worthless. Some finite difference approximations appear to be stable under any condition; others are always unstable; while some are stable only if the spacing intervals satisfy certain requirements.