• Debora AmadoriEmail author
  • Laurent Gosse
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we introduce the topic of the book, namely the class of partial differential equations on which it is focused and an Outline of the presented material.


Hyperbolic system of balance laws Accuracy of schemes for balance laws 

1.1 Some General Perspective

Even if post-WWII supersonic aircrafts development considerably stimulated both the understanding and practical approximation schemes for inviscid fluid mechanics equations, multidimensional systems of conservation laws [9], the decision to massively invest into the computational treatment of Euler equations appears to be a bit more recent, roughly speaking the 80s, when the shortcomings of popular widely-used models based on potential flow hypotheses were revealed during the so-called “Stockholm Olympics”, see [29]. A previous success for (hypersonic) computational fluid dynamics was the correct design of the thermal shield on the Apollo spatial vehicles for the atmosphere reentry phase, based on a multi-D extension of the Godunov scheme produced in 1959 for inviscid Euler equations, [14, 35]. Oppositely, during the 80s, the Airbus 320 transonic airfoils were still designed by means of a potential flow approximation, i.e. a scalar equation for which general conservation of mass, momentum and energy is not completely ensured.

1.1.1 Inviscid Systems of Conservation Laws

Obviously, an heavy price inherent to switching from the computational simulation of an elementary, possibly fitted, model to the handling of a considerably more complex system of equations is the design of reliable, realizable and efficient numerical algorithms. Presently, multidimensional inviscid Euler equations appear to be replete of difficulty of all kinds, like the mathematical ones in order to give some rigorous sense to weak discontinuous solutions containing shock waves, [5, 26], but also some unexpected ones, like several non-uniqueness and ill-posedness phenomena recently established for 2D inviscid systems containing vorticity , [8, 12]. It came with no surprise that fine-scale structures concomitant with vorticity developing in the solution may result into a destabilizing factor in numerical computations; less expected was the fact that it overcomes the stabilizing effect of entropy dissipation, even at the mathematical level of continuous equations. Paradoxically, some authors begin to take a step back and advocate some potential flow models: see [11].

Besides, left apart ill-posedness pathologies, numerical issues may pop up immediately when one plans to compute multidimensional flows endowed with discontinuous shock waves by means of finite-differences on a fixed grid: P.L. Roe calls the example of a static shear flow not aligned with the grid (i.e. an oblique contact discontinuity) the “case against upwinding”: see [19, p. 345] and Fig.  6.7. Examples of this kind sparkled many developments and algorithmic advances, some of which were consigned in the book [20], see also [36]. Even if many pathologies still remain in existing algorithms, some signs suggest that algorithmic innovation slowed down so much that it practically came to a stop, [31], the main driving force behind recent computational progress being mostly an increase in CPU power.

1.1.2 One-Dimensional Systems

Enormous difficulties coming from the multidimensional characters of compressible flows [13] can be somewhat circumvented by following Godunov’s ideas, namely narrowing the scope only to one-dimensional models within a dimensional-splitting perspective (see e.g. [17, 24]). Necessity knows no law. A salient feature in one-dimensional inviscid models is the availability of an efficient tool for the resolution of discontinuities, the Riemann problem and its “simple waves” endowed with neat directions of propagation; such a building-block gave rises to many time-marching approximation procedures, which all rely on resolving discontinuities (possibly induced through wave interactions ) by correctly distinguishing the nature of left- and right-going disturbances. A piecewise-constant approximation of initial data furnishes a convenient input allowing to set up many Riemann-based algorithms, leading to significant theoretical stability results, too [5]. Of course, one should always keep in mind that, apart from specific problems, one-dimensional models are a drastic simplification which cannot be thought as truly reflecting the complexity of certain multidimensional inviscid flows [10, 16, 27, 30, 40].

Even in such a simplified context, both theoretical and computational issues remain: left aside the scalar conservation law for which a satisfying theory was proposed by Kružkov [23] and the so-called Temple class systems [6], general 1D strictly hyperbolic systems of conservation laws are usually subject to restrictive smallness assumptions on their (initial, boundary) data [39]. In the realm of such a \(L^1\)-stability theory of (small) \({\textit{BV}}\) entropy solutions, one still faces several important obstructions when considering numerical approximation processes, except for a few notable cases, like Godunov’s scheme for Temple class systems [7] (in contrast with [2, 3]), or the wavefront tracking algorithm [28], a convenient convergence theory of widely-used schemes is still not available. Worse, obstacles appear to be not just technical, as some quite practical examples were produced, displaying for instance wave-curve deformations as a result of numerical viscosity [1, 22, 37].

1.1.3 Source Terms Inclusion

So far, only homogeneous models were tacitly considered, that is to say, models endowed with neither forcing nor dissipation, for which global conservation exactly holds along time. Needless to say, a good description of realistic problems often involves supplementary, lower-order terms, usually referred to as source terms, likely to restrict, or at least perturb a bit more existing stability theories. Two main classes of source terms can be distinguished:
  • Dissipative mechanisms, like relaxation appearing in discrete models of more complex kinetic equations, which effect is to push the system onto an equilibrium manifold where its dynamics are described by a reduced number of variables.

  • Possibly accretive terms of bounded extent thus naturally position-dependent, yielding a scattering mechanism as they are somewhat negligible at infinity, but create strong interactions with convective waves in the vicinity of the origin.

Numerical techniques for both these categories are intrinsically different: uniform in space, dissipative terms are well handled by operator-splitting in time algorithms, consisting essentially in alternating the treatment of the homogeneous equations and the ordinary differential system resulting of the presence of the source (which acts more like a sink ) [17]. Oppositely, position-dependent, bounded-extent source terms lead, in large times, to a scattering (i.e. non-interacting) state where homogeneous hyperbolic waves propagate far away, but a steady-state wave, resulting from a delicate balancing of convection forces and sources, stands close to the origin. Such a balance is usually not correctly reproduced by a time-splitting strategy, as was already well understood in the 80s [18, 34]. A scattering mechanism is present in homogeneous systems of conservation laws: by strict hyperbolicity, the solution decays toward a set of non-interacting waves. This set of waves identifies with the Riemann problem at infinity, obtained by a “zoom-out” rescaling. Yet, in presence of a bounded extent source, such a rescaling wipes off everything except for a Dirac measure in zero inside which all the effects are concentrated. A S-matrix relates incoming and outgoing hyperbolic waves, locally scattered by a “Dirac source” [13].

1.2 This Book in a Nutshell

1.2.1 Outline of the Contents

This book is mostly concerned with quantifying the accuracy of specific numerical approximations of 1D systems of balance laws endowed with position-dependent, possibly accretive source terms, hence decaying in large time onto a scattering state involving a stationary wave. These approximations follow Glimm’s canvas, namely reducing a position-dependent source term to a countable collection of “local scattering centers” by means of Dirac measures (for instance, located at the edges of computational cells in a Cartesian mesh), and proceeding by iteratively solving all the resulting Riemann problems, but this time, including the “standing wave ” (in the terminology of [21]) which results of the “Dirac source scattering ”. Within a Godunov scheme, this procedure is mostly referred to as well-balanced scheme [15] whereas in the context of homogenization of periodically forced scalar laws, one speaks more willingly about a generalized Glimm scheme [38]. During the last decade, such a numerical strategy was very successful on practical computations for shallow water equations endowed with steep topography source terms (see [4] and ( 2.11); hence the natural question “is there a rigorous mathematical explanation for this class of schemes outperforming the standard ones ?” to which the forthcoming chapters aim at (partly) answer in a unified, self-contained manner.

  • Chapter  2 deals with several forms of error estimates for 1D systems of balance laws. Especially, the classical so-called Local Truncation Error (L.T.E.) is introduced, following an analysis of K. Morton. Most “high-order schemes” for conservation laws rely on this (formal) notion of local error for their construction. Since it usually employs Taylor expansions of the solution, its relevance in the context of weak, possibly discontinuous, solutions isn’t obvious: two simple examples (the deformation of shock curves by numerical viscosity and a nonlinear shock-rarefaction interaction ) reveal some of its shortcomings. Moreover, such a local error bound should be integrated in time, with the help of convenient uniform bounds, in order to produce a global error estimate which quantifies the actual error separating the approximate solution from its exact counterpart in some norm (usually \(L^1\)): by assuming infinite smoothness for solutions of a linearized shallow water system, it is shown that handling the topography term a(x) as a collection of “local scattering centers” furnishes a much more robust approximation. This elementary calculation is checked on a practical example.

  • Chapter  3 focuses onto the 1D, possibly accretive , position-dependent scalar balance law ( 3.1). For simplicity, only one numerical algorithm is considered, the wavefront-tracking (WFT) [5, 10], for which a general \(L^1\) stability theory was established by means of a specific Lyapunov functional . Such a functional keeps track of the time-evolution of the distance between two approximate WFT solutions emerging from different \({\textit{BV}}\) initial data. It’s easy to see that under some mild hypotheses (one of these being the non-resonance assumption [21, 25]), this functional being uniformly equivalent to the \(L^1\) norm furnishes a reliable error estimate allowing to bypass the Gronwall lemma, hence free from any exponential dependence in time! Another feature of such an analysis is revealed when considering the particular case of a periodically forced scalar law ( 3.42), for which an original variant of the stability functional allows to improve significantly previous estimates. Practical tests are displayed for both cases, too.

  • Chapter  4 is probably the most ambitious; it explores the possibility to extend the good error estimates of the former one, from the scalar equation toward a semilinear, position-dependent system endowed with a relaxation-type source term. Systems of this type can be met in many areas of application, left aside well-known relaxation approximations to scalar conservation laws. Being semilinear, it allows for the derivation of an original, time-decaying, Lyapunov functional which uniformly controls the global \(L^1\) error of the numerical process. Such a functional is absolutely specific to the “local scattering centers” procedure and it’s rather easy to produce examples for which one sees without doubt that it outperforms the Kuznetsov-type estimates which govern the error of more standard discretizations. This fact is checked numerically, too, in Chap.  5.

  • Last but not least, Chap.  6 aims at giving hints about more complex models for which our analysis may be extended: such systems may include weakly nonlinear models involving a coupling with a self-consistent Poisson equation (rendering electrostatic interactions, biological confinement or gravity forces) or so-called Temple class systems met for instance in the context of traffic flow modeling. A widely open question remains 2D applications: as Chap.  4 was largely concerned with a semilinear problems being lower-order perturbations of the 1D wave equation, the Riemann problem for the 2D linear wave equation in variables puv is studied: an analytical expression of its exact solution is given and displayed on Fig.  6.5. Very complex wave interactions appear inside the Kirchhoff disc as soon as initial data are endowed with some vorticity . This is the basic building block in order to construct a genuinely 2D Godunov scheme.

1.2.2 The Ariadne’s Thread

Basically, our conclusions can be stated in a straightforward fashion as follows:
  1. 1.

    Glimm’s method of concentrating bounded extent, position-dependent source terms as “local scattering centers” arranged on a discrete lattice consists essentially in treating them by means of supplementary wave interactions .

  2. 2.

    If the source term reads k(x)g(u), it is convenient to rewrite it \(g(u)\partial _x a\), being a(x) an antiderivative of k satisfying the trivial equation \(\partial _t a=0\). One immediately deduces that ua solve an homogeneous system, with a supplementary (immobile) characteristic family, the “standing wave ” in [21] (see also [25, 33]).

  3. 3.

    Under the non-resonance assumption, such an augmented system is strictly hyperbolic hence admits a Lyapunov functional decaying along convenient pair of \({\textit{BV}}\) solutions, say ua and vb. This decay is proved without invoking the Gronwall lemma, so no time-exponentials are involved. The source term’s effects are generally handled by means of an interaction potential, a classical object in Glimm’s theory which may impose smallness restrictions, though.

  4. 4.

    Being that Lyapunov functional uniformly equivalent with the \(L^1\) norm, a global error estimate is easily deduced. However, since it depends only on the \({\textit{BV}}\) norms of the data, mechanically it perceives only the \(L^1\) norm of the position-dependent coefficient. For instance, the presence of k(x)g(u) produces an error depending only of \(\Vert k\Vert _{L^1}\), but not on any of its derivatives. This is most probably the origin of the accuracy on shallow water models endowed with a steeply varying topography term: the global error are insensitive to its oscillations (such a remark is relevant in homogenization of oscillating balance laws [38], too).

  5. 5.

    The non-resonance assumption is not to be taken lightly: a formal analysis of constants showing up in the error estimate for balance laws reveals a term like \(|g(u)/f'(u)|\), being \(f'(u)\) the velocity field. Such a quantity blows up if the augmented system ceases to be strictly hyperbolic [21], so one must expect the accuracy to decrease in the vicinity of sonic or stagnation points. See for instance Fig.  3.3, where an “error spike” is located at a point where \(f'(u)\simeq 0\).



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© The Author(s) 2015

Authors and Affiliations

  1. 1.DISIMUniversità degli Studi dell’AquilaL’AquilaItaly
  2. 2.Istituto per le Applicazioni del CalcoloCNRRomeItaly

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