  The usual explicit finite-difference method of solving partial differential
equations is limited in stability because it approximates the exact
amplification factor by power-series. By adapting the same
exponential-splitting method of deriving symplectic integrators, explicit
symplectic finite-difference methods produce Saul'yev-type schemes which
approximate the exact amplification factor by rational-functions. As with
conventional symplectic integrators, these symplectic finite-difference
algorithms preserve important qualitative features of the exact solution. Thus
the symplectic diffusing algorithm is {\it unconditionally stable} and the
symplectic advection algorithm is {\it unitary}. There is a one-to-one
correspondence between symplectic integrators and symplectic finite-difference
methods, including the key idea that one can systematically improve an
algorithm by matching its modified Hamiltonian more closely to the original
Hamiltonian. Consequently, the entire arsenal of symplectic integrators can be
used to produce arbitrary high order time-marching algorithms for solving the
diffusion and the advection equation.
