A gravity-wave problem with the upstream difference method

In numerous applications, upstream differencing has been used to treat the horizontal advection terms in primitive-equation models of the atmosphere. Though solutions to these equations are characterized by the presence of gravity waves, the velocity of these waves is not taken into account in formulating the difference equations. Determination of the “upstream” direction is based upon consideration of the transport velocity alone. With this approach, very small time steps are required to maintain computational stability when the gravity wave speed is large compared to the advecting speed. For the modeler who wishes to retain upstream differencing in such cases, a simple method is presented to enlarge the stability region.