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.
