Positive Finite-Difference Advection Scheme Applied on Locally Refined Grids

摘要

A class of explicit finite difference advection schemes derived along the methodof lines is examined. An important problem field is large scale atmospheric transport, and therefore focus is on the demand of positivity and on the use of locally refined grids. For the spatial discretization, attention is confined to directionally split schemes in conservation form using five points per direction. The fourth order central scheme and the family of k-schemes comprising the second order central, the second order upwind, and the third order upwind biased discretization are studied. Positivity is enforced through flux limiting. The limited third order upwind discretization is concluded to be the best candidate from the four examined. For the time integration, attention is confined to a number of explicit Runge-Kutta methods of orders two up to four. With regard to the demand of positivity, these integration methods turn out to behave almost equally and no best method could be identified. For solutions with steep gradients, a local uniform grid refinement technique is proposed as an effective tool to speed up computations.