Computing Qualitatively Correct Approximations of Balance Laws: Exponential-Fit, Well-Balanced and Asymptotic-Preserving (SEMA SIMAI Springer Series Book 2)

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. La 4ème de couverture indique : 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 of 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 Front Matter....Pages i-xix Introduction and Chronological Perspective....Pages 1-17 Front Matter....Pages 19-20 Lifting a Non-Resonant Scalar Balance Law....Pages 21-40 Lyapunov Functional for Linear Error Estimates....Pages 41-61 Early Well-Balanced Derivations for Various Systems....Pages 63-76 Viscosity Solutions and Large-Time Behavior for Non-Resonant Balance Laws....Pages 77-93 Kinetic Scheme with Reflections and Linear Geometric Optics....Pages 95-116 Material Variables, Strings and Infinite Domains....Pages 117-134 Front Matter....Pages 135-136 The Special Case of 2-Velocity Kinetic Models....Pages 137-165 Elementary Solutions and Analytical Discrete-Ordinates for Radiative Transfer....Pages 167-189 Aggregation Phenomena with Kinetic Models of Chemotaxis Dynamics....Pages 191-214 Time-Stabilization on Flat Currents with Non-Degenerate Boltzmann-Poisson Models....Pages 215-239 Klein-Kramers Equation and Burgers/Fokker-Planck Model of Spray....Pages 241-261 A Model for Scattering of Forward-Peaked Beams....Pages 263-268 Linearized BGK Model of Heat Transfer....Pages 269-293 Balances in Two Dimensions: Kinetic Semiconductor Equations Again....Pages 295-313 Conclusion: Outlook and Shortcomings....Pages 315-321 Back Matter....Pages 323-341