On the Cancellation Problem in Calculating Compressible Low Mach Number Flows

Abstract

In calculating low Mach number flows one faces the stiffness problem in two different facets:•the applicable time steps become very small•the constants of the cancellation errors become very large (1/γ).Usually the first point receives attention. Here we want toconcentrate on the cancellation problem only.To our knowledge there is no detailed investigation of this problem in the related literature. In primitive variable formulations the problem can be solved by using the pressure coefficient instead of the pressure and a similar variable for the temperature or the internal energy. In conservative variable formulations this is thought not to be possible and therefore is sacrificed. We are able to show that a local reference state can also be introduced into a conservative scheme, if carefully applied to all quantities and applied to all constituent parts of the program. A detailed error analysis is given for all these parts. Finally, we show that we can perform a very low Mach number calculation atM=10−11with a seven digits arithmetic only and still maintain the set of conservative variables. The governing equations are unaltered and the method depends neither on the time integration scheme nor the specific discretization. The method should be used in connection with the standard strategies like preconditioning, multigrid, or an (semi-)implicit method if acceleration is desired.