Insights from a pseudospectral approach to the Elder problem

Abstract

[1] The aim of this paper is to clarify and circumvent the issue of multiple steady state solutions in the Elder problem. A pseudospectral method is used to avoid numerical error associated with spatial discretization. The pseudospectral method is verified by comparison to an analytical solution at Rayleigh number, Ra = 0, and by reproducing the three stable steady state solutions that are known to exist at Ra = 400. A bifurcation diagram for 0 < Ra < 400, which is free of discretization error, confirms that multiple steady states are indeed an intrinsic characteristic of the Elder problem. The existence of multiple steady states makes the Ra = 400 Elder problem less suitable for benchmarking numerical models. To avoid the multiple steady states, we propose a benchmark at Ra = 60. The results for this Low Rayleigh Number Elder Problem are presented and compared to simulations with the commercial groundwater modeling package FEFLOW. Correspondence between the pseudospectral model and FEFLOW is excellent.

The Elder Problem

“…Secondly, whilst Frolkovič and De Schepper (2001) [3] report 3 stable steady-state cases they made no attempt to quantify the bifurcation diagram and to determine the existence of these three stable steady-states for different Rayleigh numbers. This would, however, soon be the subject of work by Johannsen (2002Johannsen ( , 2003 [5,6] and then later van Reeuwijk et al (2009) [7]. Figure 5.…”

“…However, notably the studies of Frolkovič and De Schepper (2001) [3], Johannsen (2002Johannsen ( , 2003 [5,6] and van Reeuwijk et al (2009) [7] have resolved the confusion by showing multiple states are a reality for Here is a little bit of history. We start at the very beginning with Elder's early years; describe the motivation for the study of the natural convection as part of his work at Cambridge; examine the uptake of Elder's Hele-Shaw cell and numerical experiments into modern fluid mechanics and modelling; and finally describe the current understanding of the Elder Problem-including the most recent rigorous bifurcation study using the pseudospectral method that has helped to finally resolve the confusion.…”

“…Several weeks later by email they started to discuss A benchmark needs to be reliable and not clouded in uncertainties. To see if the Elder Problem does not suffer from these uncertainties, van Reeuwijk, Mathias, Simmons and Ward (2009) [7] employed the pseudospectral method particularly to avoid numerical error associated with mesh spacing. Results confirmed that multiple steady states (rather than numerically spurious states) are indeed real -intrinsic characteristics of the Elder Problem.…”

“…Results confirmed that multiple steady states (rather than numerically spurious states) are indeed real -intrinsic characteristics of the Elder Problem. van Reeuwijk et al (2009) [7] produced the bifurcation diagram for the Elder Problem for 0 < Ra < 400 by plotting Sherwood number (Sh) versus Rayleigh number (Ra). They observed: "The only solution which exists for the entire range of Ra is S 1 .…”

Simulation of Three-Component Two-Phase Flow in Porous Media Using Method of Lines

“…The spatial dimension is typically treated using conservative methods such as finite volume (Chen et al 2006, p. 128). Alternatively, one can consider the use of finite elements (Chen et al 2006, p. 94) or pseudospectral methods (van Reeuwijk et al 2009). Such spatial schemes give rise to either stability problems or numerical diffusion due to truncation terms associated with the Taylor's expansion, the latter of which can be reduced using flux limiters or their variants (e.g., Mallison et al 2005).…”