Finite difference solutions for internal waves in enclosures

Finite difference approximations to the set of partial differential equations governing internal waves are investigated. Analytical solutions describing waves in an enclosure in two and three dimensions are obtained. The schemes considered are second order accurate in space and include first order explicit and second order time differencing. The solutions are used to investigate the temporal stability and long term accuracy of all schemes. The mode frequencies and wave shapes obtained from each difference scheme are compared with the solutions both to the corresponding partial differential equations and to equations obtained by discretizing in space only. The solutions have been used by the authors to help develop a finite difference code designed to compute nonlinear buoyancy driven flows of the type that arise in enclosure fires.