Non-linear waves of the steady natural convection in a vertical fluid layer: A numerical approach Original Research Article

A fully implicit finite difference scheme is developed for the stream function formulation of unsteady thermoconvective flows. An artificial time is added in the equation for the stream function rendering it into an ultra-parabolic type. For each time stage of the real time, a convergence is obtained with respect to artificial time (“internal iterations”). An implicit efficient operator-splitting time stepping is designed and proved to be absolutely stable. Employing a conservative central-difference approximation of the non-linear terms makes the scheme absolutely stable without using upwind differences. As a result, the scheme has no scheme viscosity and has virtually negligible phase error, which makes it a useful tool for investigating the intricate structure of the thermoconvective flow. The scheme is second-order approximation both in time and space.