A survey of numerical techniques for solving singularly perturbed ordinary differential equations

A deferred correction method for the numerical solution of nonlinear two-point boundary value problems has been derived and analyzed in two recent papers by the first author. The method is based on mono-implicit Runge–Kutta formulas and is specially designed to deal efficiently with problems whose solutions contain nonsmooth parts—in particular, singular perturbation problems of boundary layer or turning point type. This paper briefly describes an implementation of the method and gives the results of extensive numerical testing on a set of nonlinear problems that includes both smooth and increasingly stiff (and difficult) problems. Results on the test set are also given using the available codes COLSYS and COLNEW. Although the intent is not to make a formal comparison, the code described appears to be competitive in speed and storage requirements on these problems.

A Deferred Correction Method for Nonlinear Two-Point Boundary Value Problems: Implementation and Numerical Evaluation.

Investigation of the linear stability problem for rapidly rotating convection on an f-plane has revealed the existence of two distinct scales in the vertical structure of the critical eigenfunctions: a small length scale whose vertical wavenumber kz is comparable with the large horizontal wavenumber k⊥ selected at onset, and a large-scale modulation which forms an envelope on the order of the layer depth d. The small-scale structure in the vertical results from a geostrophic balance imposed by the Taylor–Proudman constraint. This primary balance forces rotational alignment and confines fluid motions to planes perpendicular to the rotation axis. For convective transport in the vertical this constraint must be relaxed. This is achieved by molecular dissipation which allows weak upward (downward) spiralling of hot (cold) fluid elements across the Taylor–Proudman planes and results in a large-scale vertical modulation of the Taylor columns.In the limit of fast rotation (i.e. large Taylor number) a multiple-scales analysis leads to the determination of a critical Rayleigh number as a function of wavenumber, roll orientation and the tilt angle of the f-plane. The corresponding critical eigenfunction represents the core solution; matching to passive Ekman boundary layers is required for a complete solution satisfying boundary conditions.An extension of this analysis, introduced by Bassom & Zhang (1994), is used to describe strongly nonlinear two-dimensional convection, characterized by significant departures of the mean thermal field from its conduction profile. The analysis requires the solution of a nonlinear eigenvalue problem for the Nusselt number (for steady convection) and the Nusselt number and oscillation frequency (for the overstable problem). The solutions of this problem are used to calculate horizontal and vertical heat fluxes, as well as Reynolds stresses, as functions of both the latitude and roll orientation in the horizontal, and these are used to calculate self-consistently north–south and east–west mean flows. These analytical predictions are in good agreement with the results of three-dimensional simulations reported by Hathaway & Somerville (1983).

Strongly nonlinear convection cells in a rapidly rotating fluid layer: the tilted f-plane

It is a well established result of linear theory that the influence of differing mechanical boundary conditions, i.e., stress-free or no-slip, on the primary instability in rotating convection becomes asymptotically small in the limit of rapid rotation. This is accounted for by the diminishing impact of the viscous stresses exerted within Ekman boundary layers and the associated vertical momentum transport by Ekman pumping. By contrast, in the nonlinear regime recent experiments and supporting simulations are now providing evidence that the efficiency of heat transport remains strongly influenced by Ekman pumping in the rapidly rotating limit. In this paper, a reduced model is developed for the case of low Rossby number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominately aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need for spatially resolving the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer the nonlinear feedback from Ekman pumping would be able to generate a heat transport that diverges to infinity. This layer arrests this blowup resulting in finite heat transport at a significantly enhanced value.

A nonlinear model for rotationally constrained convection with Ekman pumping

/pdf/almost-block-diagonal-linear-systems-sequential-and-parallel-414641hiek.pdf

Almost block diagonal linear systems: sequential and parallel solution techniques, and applications

In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a “one-off” formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.

High order methods for the numerical solution of two-point boundary value problems

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High order methods for the numerical solution of two-point boundary value problems

High order methods for the numerical solution of two-point boundary value problems