Understanding Saul'yev‐type unconditionally stable schemes from exponential splitting

Saul'yev‐type asymmetric schemes have been widely used in solving diffusion and advection equations. In this work, we show that Saul'yev‐type schemes can be derived from the exponential splitting of the semidiscretized equation which fundamentally explains their unconditional stability. Furthermore, we show that optimal schemes are obtained by forcing each scheme's amplification factor to match that of the exact amplification factor. A new second‐order explicit scheme is found for solving the advection equation with the identical amplification factor as the implicit Crank–Nicolson algorithm. Other new schemes for solving the advection–diffusion equation are also derived.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1961–1983, 2014