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## Über dieses Buch

Substantial effort has been drawn for years onto the development of (possibly high-order) numerical techniques for the scalar homogeneous conservation law, an equation which is strongly dissipative in L1 thanks to shock wave formation. Such a dissipation property is generally lost when considering hyperbolic systems of conservation laws, or simply inhomogeneous scalar balance laws involving accretive or space-dependent source terms, because of complex wave interactions. An overall weaker dissipation can reveal intrinsic numerical weaknesses through specific nonlinear mechanisms: Hugoniot curves being deformed by local averaging steps in Godunov-type schemes, low-order errors propagating along expanding characteristics after having hit a discontinuity, exponential amplification of truncation errors in the presence of accretive source terms... This book aims at presenting rigorous derivations of different, sometimes called well-balanced, numerical schemes which succeed in reconciling high accuracy with a stronger robustness even in the aforementioned accretive contexts. It is divided into two parts: one dealing with hyperbolic systems of balance laws, such as arising from quasi-one dimensional nozzle flow computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes systems. Stability results for viscosity solutions of onedimensional balance laws are sketched. The other being entirely devoted to the treatment of weakly nonlinear kinetic equations in the discrete ordinate approximation, such as the ones of radiative transfer, chemotaxis dynamics, semiconductor conduction, spray dynamics or linearized Boltzmann models. “Caseology” is one of the main techniques used in these derivations. Lagrangian techniques for filtration equations are evoked too. Two-dimensional methods are studied in the context of non-degenerate semiconductor models.

## Inhaltsverzeichnis

### Chapter 1. Introduction and Chronological Perspective

Abstract
This introductory chapter aims at positioning the book’s primary topics according to both a scientific and an historic context; loosely speaking, the objective here is more trying to unify seemingly different sectors in numerical analysis rather than being very specific (this will come later on). In particular, one can figure out the main ideas exposed in the sequel by examining very classical computations which trace back to 1960–70, namely the passage from finite differences to exponentially-fitted schemes for transient convection-diffusion equations. In some sense, well-balanced schemes are but an extension of these methods for hyperbolic problems: the link being provided by both the finite volumes discretization (what was formerly called the “box scheme”) and the exact solving of the steady-state equations in order to compute the numerical fluxes at each interface of the computational grid. A point of crucial importance is the following (quoting [61, p. 159]):
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### Chapter 2. Lifting a Non-Resonant Scalar Balance Law

Abstract
The mathematical theory of scalar conservation laws has reached a state of completion: existence, uniqueness, regularity and stability with respect to initial data have been established in various settings (BV theory, compensated compactness, kinetic formulation, relaxation approximation, etc…). Here, we aim at presenting the special features holding when space-dependent, non-dissipative, source terms are added on the right-hand side (which complicates the picture), but under the simplifying assumption that stagnation points aren’t allowed (f′(u) ≠ 0).. This somehow tempers the effects of the source, for instance when it has compact support in ℝ, allowing for the derivation of peculiar, uniform in time, bounds in both amplitude and total variation which express the fact that convective waves exit the amplification area after some time. Such bounds lead to an improvement in Kuznetsov-type error estimates [24] for the so-called well-balanced schemes, obtained by approximating a 2 × 2 homogeneous Temple-class system by means of Godunov’s method.
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### Chapter 3. Lyapunov Functional for Linear Error Estimates

Abstract
The main goal of the present Chapter is to emphasize the qualitative difference between Time-Splitting (TS, also called Fractional Step, FS) and Well-Balanced (WB) numerical schemes when it comes to computing the entropy solution [26] of a simple scalar, yet non-resonant, balance law:
$${\partial}_tu+{\partial}_xf(u)=k(x)g(u),\kern1.68em 0\le k\in {L}^1\cap {C}^0\left(\mathbb{R}\right)$$
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### Chapter 4. Early Well-Balanced Derivations for Various Systems

Abstract
Even more than the formers ones, the present chapter will emphasize that well-balanced methods consist essentially in recycling astutely homogeneous techniques in order to take advantage of their strong stability properties in the more delicate context of non-homogeneous systems: for instance, the well-known fact that the Godunov scheme has zero numerical viscosity at steady-state (the preservation of steady-state curves is actually a consequence of this feature). Obviously, there is a price to pay: if nearly all source terms can be reformulated painlessly as nonconservative products, the corresponding linearly degenerate fields can locally interfere with the already existing characteristic fields of the original equations, meaning that strict hyperbolicity doesn’t hold unconditionally. Such a pathological situation is referred to as non-linear resonance and led to numerous theoretical investigations. Kinetic formalism can sometimes save the day: in Chapter 6, we shall study a kinetic scheme which doesn’t suffer from any resonance issue. Hereafter, nonlinear resonance will be avoided, following the assumptions of the seminal paper [39].
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### Chapter 5. Viscosity Solutions and Large-Time Behavior for Non-Resonant Balance Laws

Abstract
Our aim is now to summarize several rigorous results concerning BV solutions of general hyperbolic systems of balance laws with source terms about which no dissipation hypothesis is made. Results of this kind owe to the pioneering 1979 paper by Tai-Ping Liu [38], already quoted in Chapter 4, inside which an astute extension of the Glimm scheme [21] is studied in great detail. His existence theorems can be somewhat simplified when source terms are concentrated inside the whole Riemann fan instead of being localized in the middle of computational cells: hence the presentation adopted in [1] will be preferred. Uniqueness results were also proved in that more recent paper (a slightly less complete result was obtained in [24]). A second part in the chapter focuses onto decay results: first, positive (rarefaction) waves are handled following the paper [23]. Second, weak and strong decay results dealing with time-asymptotic behavior are recalled from [38]: as a consequence of the interaction potential’s decay, the BV solution endowed with constant states at infinity evolves toward a non-interacting wave pattern, which resembles more and more to the fan associated to the Riemann problem resolving that discontinuity. Inside the area |x| ≤ M where the source term is powerful, a steady-state profile, smooth solution of ∂ x f (u) = g(x, u) emerges as time passes whereas in the complementary domain, only homogeneous hyperbolic waves survive. In other words, this is exactly the setup for scattering theory, namely a wave phenomenon studied in a domain which is large compared to the characteristic scale of the interaction. The non-resonance assumption |∇f (u)| ≠ 0 is of paramount importance here because it ensures that convective waves never stagnate and exit the interval |x| ≤ M in finite time thus the solution cannot be amplified in an unbounded manner by the source term.
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### Chapter 6. Kinetic Scheme with Reflections and Linear Geometric Optics

Abstract
In [37], Perthame and Simeoni proposed a well-balanced scheme for shallow water equations which main core consists in solving a modified Vlasov equation,
$${\partial}_tf+v{\partial}_xf-{\partial}_xV(x)\;{\partial}_vf={\displaystyle \sum_{n\in \mathbb{N}}\left(M\left(\rho, v-u\right)-f\right)\delta \left(t- n\varDelta t\right),}$$
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### Chapter 7. Material Variables, Strings and Infinite Domains

Abstract
Within certain applications, one faces a differential model which is to be solved for instance on the totality of the real line ℝ, with convenient decay properties at infinity. This can happen for quantum models, where the characteristic scale is small so that a component of macroscopic length becomes infinite. It also arises on the other side of the spectrum, namely for cosmological models where one seeks to visualize gravitational effects of primordial matter fluctuations (Jeans instability, stars collapse and galaxy formation). A last example finds itself in the study of seismic wave propagation in elastic media or the oceanic models [23, 31]. For linear wave models, several remedies exist: among them, the use of Fourier techniques [2] or the design of appropriate boundary conditions at the edges of a limited computational domain [18, 23, 31]. When the differential model becomes nonlinear, the necessity of simulating on a whole infinite domain may arise from peculiar behavior of the solution, like the decay toward intermediate asymptotics which can be observed only after merging of all connected components [3, 13]. For conservative one-dimensional models, the introduction of the Lagrangian “material variable” generates a formalism for which no difficulty remains when it comes to carry out computations on the whole real line. As it contains the differential equations of motion, $$\dot{X}=U,\dot{U}=-\phi \hbox{'}(X)$$, the Lagrangian numerical scheme displays interesting features, like for instance a contraction property in the Wasserstein metric in the context of filtration equations (which ensures that edges propagate at the correct velocity in the degenerate case). This decay of a specific nonlinear metric may be related with the one studied in Chapter 3 with Eulerian well-balanced schemes.
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### Chapter 8. The Special Case of 2-Velocity Kinetic Models

Abstract
This chapter deals mainly with the numerical analysis of the following one-dimensional system of semilinear equations,
$$\begin{array}{ccc} {\partial}_t{f}^{\pm}\pm {\partial}_x{f}^{\pm }=G\left({f}^{+},{f}^{-}\right), & x\in \mathbb{R}, & t>0, \end{array}$$
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### Chapter 9. Elementary Solutions and Analytical Discrete-Ordinates for Radiative Transfer

Abstract
This chapter is entirely devoted to the exposition of the method of elementary solutions, which has been introduced and developed during the 50’s–60’s mainly by Chandrasekhar, Case, Cercignani, Siewert and Zweifel. In particular, Chandrasekhar’s discrete ordinates approximation has been refined by Siewert and his collaborators into a so-called analytical discrete ordinates (ADO) method through a systematic use of elementary solutions. It appeared that it was exactly what is needed in order to set up a time-dependent well-balanced numerical scheme for linear kinetic equations by furnishing an explicit scattering matrix at each interface of the computational grid. The main goal is first to derive a WB scheme which solves the Cauchy problem for a simple model of “grey” radiative transfer:
$$\begin{array}{cc} {\partial}_tf+v{\partial}_xf=\frac{c}{2}{\displaystyle {\int}_{-1}^1f\left(t,x,{v}^{\prime}\right)d{v}^{\prime }-f,} & v\in \left[-1,1\right],x\in \mathbb{R},t>0. \end{array}$$
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### Chapter 10. Aggregation Phenomena with Kinetic Models of Chemotaxis Dynamics

Abstract
Chemotaxis describes the directed movement of a large population of micro-organisms (cells, bacteria, …) in response to a gradient of a chemical substance, usually referred to as the chemo-attractant. Indeed the swimming displacement of certain flagellated bacteria can be described by straight-line “runs” suddenly interrupted by “tumbles” of very short duration (see realistic time-scale measures in [19, Table 1.1], and [43]). Usually, cell velocities are of the order of 10/20 μm/sec, run lengths can be of 10/20 cell diameters and tumbling lasts at most 0.1 sec. In the absence of chemo-attractant, their motion would be a random walk; however, its presence results in a decrease of tumbling frequency when cells move in the direction of increasing concentration, a process called chemokinesis which tends to increase run lengths in the favorable direction.
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### Chapter 11. Time-Stabilization on Flat Currents with Non-Degenerate Boltzmann-Poisson Models

Abstract
Electronic transport can be studied within the framework of kinetic theory, being itself closely related to homogenization limits of quantum models, cf. e.g. [7, 38], because it deals with a statistical description which makes sense in view of the many electrons in a typical semiconductor. Accordingly, one deals with f (t, x, k) ∈ [0,1], a distribution function describing the statistical repartition of electrons at time t ≥ 0, located around the location x ∈ ℝ3 and with a wave vector k ∈ ℬ ⊂ ℝ3, the so-called first Brillouin zone [62]. Any such (semiclassical) particle has an energy ℰ(k) given by a smooth dispersion relation; its velocity v(k) reads:
$$\mathbf{v}\left(\mathbf{k}\right)=\frac{1}{\hslash}\;{\nabla}_{\mathbf{k}}\mathcal{E}\left(\mathbf{k}\right)$$
.
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### Chapter 12. Klein-Kramers Equation and Burgers/Fokker-Planck Model of Spray

Abstract
In the preceding chapter, it has been shown that the inclusion of a Vlasov-type acceleration term inside the framework of well-balanced schemes for linear relaxation kinetic models leads to complications. There is an alternative: namely, when considering a Fokker-Planck approximation of the relaxation term, the steady-state equation can be reduced to a Sturm-Liouville eigenvalue problem. Techniques available for this class of differential equations allow for a nearly complete treatment and the spectral technique of “elementary solutions” can be set up in order to produce well- balanced schemes for which the CFL condition is affected neither by the Vlasov term, nor by the drift-diffusion term in the v variable.
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### Chapter 13. A Model for Scattering of Forward-Peaked Beams

Abstract
This shorter chapter is devoted to the numerical study of a particular Fokker-Planck equation (also called Vlasov-Lorentz model in [7]),
$${\partial}_tf+v{\partial}_xf=\sigma \partial v\left(\left(1-{v}^2\right){\partial}_vf\right),f\left(t=0,x,v\right)={f}_0\left(x,v\right).$$
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### Chapter 14. Linearized BGK Model of Heat Transfer

Abstract
In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro -differential equation whereas the heat transfer is described by a 2 × 2 coupled system.
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### Chapter 15. Balances in Two Dimensions: Kinetic Semiconductor Equations Again

Abstract
The equations studied in Chapters 11 and 12, Vlasov-BGK and Vlasov-Fokker-Planck, are genuinely bi-dimensional problems. Thanks to their special structure, one can succeed in solving them by means of essentially one-dimensional algorithms because the formalism of elementary solutions can be extended up to some numerically tolerable approximations. Besides, it is somewhat tacitly assumed that one of the 2 directions of propagation is dominant. This leaves open the possibility of attacking these problems by means of truly bi-dimensional numerical schemes, treating the Vlasov acceleration term through a divided difference in the v direction, itself possibly containing a modified state which renders locally a source term’s effect. Kinetic problems are well suited for an investigation of bi-dimensional well-balanced discretizations also because the passage from one- to two-dimensional upwind schemes is generally associated to a change of paradigm: one switches from a nonlinear flux term like x f(u) to a linear advection equation t u + a∂ x u + b∂ y u = 0. Kinetic equations, like Vlasov equation (6.2), can be seen as being in midstream.
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### Chapter 16. Conclusion: Outlook and Shortcomings

Abstract
The so-called method of artificial viscosity has been introduced in the seminal 1950 paper by Richtmyer and Von Neumann [24], where a Lagrangian hyperbolic system of gas dynamics is approximated by finite differences on staggered grids (the specific volume and the velocity aren’t known at the same points). In order to stabilize the Fourier modes of the numerical solution, it appeared necessary to include an artificial dissipative term in the pressure law, negligible in smooth areas and $$\mathcal{O}$$(1) in the vicinity of shocks. On the contrary [11], scientists in the Soviet Union
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### Backmatter

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