Eliminating grid-orientation errors in alternating-direction implicit schemes

Abstract

Many important physical problems are governed by systems of nonlinear time-dependent partial differential equations whose solutions vary on disparate time scales. Fast-scale motions are transient, and in numerical simulations one is often not interested in resolving them. Most practical implicit schemes for these problems owe their efficiency to the use of alternating-direction techniques, avoiding the solution of large systems. However, these techniques also introduce spatial factorization errors which limit the time step for which accurate results are possible. We introduce an iterative procedure to eliminate this factorization error. When it converges, the resulting scheme is equivalent to a Crank-Nicolson scheme, but it does not require the solution of large algebraic systems. We show results for the shallow water equations on a sphere, where a 90-minute time step yields accurate 48-hour results for meteorological initial data.