  We present a class of new explicit and stable numerical algorithms to solve
the spatially discretized linear heat or diffusion equation. After discretizing
the space and the time variables like conventional finite difference methods,
we do not approximate the time derivatives by finite differences, but use
constant neighbor and linear neighbour approximations to decouple the ordinary
differential equations and solve them analytically. During this process, the
timestep-size appears not in polynomial, but in exponential form with negative
exponents, which guarantees stability. We compare the performance of the new
methods with analytical and numerical solutions. According to our results, the
methods are first and second order in time and can be much faster than the
commonly used explicit or implicit methods, especially in the case of extremely
large stiff systems.
