Short noteSuitable initial conditions
Introduction
Recent discussions with distinguished specialists of numerical fluid mechanics have shown that it would be useful to write a physicist friendly version of the article [14], and this is the aim of this note.
Consider a very simple problem, namely the heat equation in space dimension one:For u0 and f given sufficiently regular, “explicit” forms of the solution u of (1) are available using the Green function of the heat equation [2]. Also mathematical results of existence and uniqueness of solutions of (1) are available when f and u0 are just square integrable, or even less regular [12]. The issues discussed in [14] do not relate to unsmooth data but rather to smooth ones. Assume for instance that f ≡ 0, and u0(x) = 1, ∀x ∈ (0, 1). The existence and uniqueness of solution of (1) is then guaranteed by many theorems, and the solution will be except near t = 0. The existence of a discontinuity near t = 0 can be seen by just observing that, for f and u0 as above, u(0, t) = 0, for all t > 0, whereas u(0, 0) = u0(0) = 1. In fact, assuming that f and u0 are , the solution u of (1) will not, in general, be smooth near t = 0. It will be so only if (when) u0 and f satisfy a sequence of conditions called the compatibility conditions, the level of regularity near t = 0 depending on the number of compatibility conditions that are satisfied. In Section 2 we describe the first and second compatibility conditions (CC) for (1), as well as for the convection and wave equations in space dimension one (the first one for (1) being u0(0) = u0(1) = 0). As another illustration of the results in [14] we present in Section 3, the substantially more complex case of the Navier–Stokes equations.
The problem of the compatibility conditions has been known and addressed in the mathematical literature for a long time; see e.g. [13] and the references in [14]. The novelty in [14] was to derive all the necessary and sufficient conditions for the solutions of certain classes of parabolic equations to be near t = 0, and especially the incompressible Navier–Stokes equations.
How this issue relates to computation? From the physical (and somehow “philosophical”) point of view, except when considering ab initium problems, any phenomenon considered for t > 0 will just be the continuation of a phenomenon which pre-existed, so that we should in principle be able to solve the problem under consideration backward in time.1 Now, for an equation like (1), given f smooth for all , the u0 for which (1) can be solved backward are relatively very rare and, for this to be true some (but not all) of the requirements on u0 are precisely the compatibility conditions. Hence solving (1) with an u0 which is not physically suitable in this sense, means, despite the beautiful mathematical theorems, that we are trying to solve this problem with a non-physical initial data. There is likely a computational price to pay for that, which is negligible for (1), but is not for more complex equations. For instance those practicing large simulations for the Navier–Stokes equations or geophysical flows, know very well that they have to “prepare” their initial data before launching the actual computations. One may wonder if this “preparation” is not related to making the initial data “suitable” in the sense above. This article does not provide any recipe but, hopefully, by shedding some light on this difficulty, may help the practitioner.
The problem of the choice of initial (and boundary) data has been discussed from many angles, in the applied and computational literature, the compatibility conditions being implicitly or explicitly mentioned. For the Navier–Stokes equations (NSE) the problem of the first and second compatibility conditions has been addressed e.g. by Heywood [9], and Heywood and Rannacher [10]; in [10] the authors emphasize the computational relevance of the compatibility conditions. In his nice book [3], Gallavotti mentions the numerical difficulty caused by an inconsistency in the initial conditions for the NS equations; this difficulty relates to the second compatibility condition (19) below, although the compatibility conditions are not alluded to. See also the book of Kreisz and Lorenz [11] who address related issues (in particular in Chapter 10, Section 10.3.2) in the context of the initialization by “the bounded derivative principle”. Several articles of Gresho alone or with co-authors address the initial and boundary conditions issues; see e.g. [4] where the first and second compatibility conditions appear; see also the review articles [5], [6]. These issues appear also explicitly in work by Boyd and Flyer [1], Flyer and Fernberg [7], and Flyer and Swarztrauber [8]; see also the references in these articles. These articles emphasize the computational impact of the CC and propose a number of remedies (in particular computing analytically the singularities near t = 0, and “removing” them from the solutions).
Section snippets
One-dimensional equations
As indicated before we now describe the first and second compatibility conditions for three very simple equations in space dimension one, the first ones being that the initial data satisfies the boundary conditions of the problem, as we already observed in the case of the heat equation.
Generalization: Navier–Stokes equations
In this section, we present the general form of the compatibility conditions and then consider the special case of the Navier–Stokes equations.
Remarks and conclusions
- (i)
The higher order compatibility conditions for the Navier–Stokes equations appear in [14] (in an abstract (mathematical) form).
- (ii)
The difficulty described here has nothing to do with the question of occurrence of singularities in finite time for the 3D Navier–Stokes equation. Our analysis relates to smooth data and solutions that are smooth for t > 0.
- (iii)
Returning to the computational problem, the unavoidable issue is, for a general u0 not satisfying (19), to solve an exact or approximate form of the
Acknowledgements
The author thanks an anonymous referee for many useful remarks and for bringing several relevant references to his attention and Qingshan Chen for providing Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6. This work was partially supported by the National Science Foundation under the grant NSF-DMS-0305110, and by the Research Fund of Indiana University.
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