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Despite all advances in multi-dimensional hydrodynamics, investigations ofstellar evolution and stellar pulsations still depend on one-dimensionalcomputations. The present work devises an alternative to the mixing lengththeory or turbulence models usually adopted for the modelling of convectivetransport in such studies. Assuming that the largest convective patternsgenerate the majority of convective transport, the convective velocity field isdescribed using two parallel radial columns to represent up- and downstreamflows. Horizontal exchange in the form of fluid flow and radiation over theirconnecting interface couples the two columns and allows a simple circulatingmotion. The main parameters of this convective description have astraightforward geometrical meaning, namely the diameter of the columns(corresponding to the size of the convective cells) and the ratio of crosssection between up- and downdrafts. For this geometrical setup, thetime-dependent solution of the equations of radiation hydrodynamics is computedfrom an implicit scheme which has the advantage of being not affected by theCourant-Friedrichs-Lewy time step limit. In order to demonstrate the approach,results for the example of convection zones in Cepheids are presented.

A two-columns formalism for time-dependent modelling of stellar convection. I. Description of the method