High order finite difference scheme for the plane problem of convection in a porous medium

Modeling of convection in a porous medium saturated with incompressible fluid has applications in geophysics, technics, etc. The problem of convection in a porous medium has a feature that distinguishes it from the problem of free fluid convection. Discovered by D. V. Lyubimov branching off a family of stationary solutions for the plane Darcy convection [3] was explained by V. I. Yudovich based on the theory of cosymmetry [4]. Galerkin method was applied to study one-parameter families of stationary states in the Darcy problem for rectangles in [5]. Second-order finite-difference scheme ensuring cosymmetry for discrete analogue was presented in [6]. The problem of convection of a heat-conducting fluid in a rectangular porous region heated from below is under consideration. We introduce a compact finite-difference scheme to supply a higher order of accuracy in solving the equations for the stream function and temperature (Darcy model). Previously, various versions of effective compact schemes for parabolic and hyperbolic equations have been developed [7–9]. We built a compact finite difference approximation on nine-point stencil by using the balance method. The constructed scheme ensures cosymmetry for discrete analogues. In a numerical experiment, we study the spectral problem and calculate critical Raleigh numbers corresponding to the branching off a family of stationary convective flows. Reducing terms in Taylor series expansion, we improved the scheme by choosing parameters in the approximation of the Laplace operator and the first derivative in horizontal coordinate. Our results show that the accuracy can be increased by choosing an optimal ratio of the step along horizontal and vertical coordinates. New finite-difference scheme allows to conduct computations of the family of steady states and its transformations more effectively