This book is devoted to the numerical analysis of compressible fluids in the spirit of the celebrated Lax equivalence theorem. The text is aimed at graduate students in mathematics and fluid dynamics, researchers in applied mathematics, numerical analysis and scientific computing, and engineers and physicists. The book contains original theoretical material based on a new approach to generalized solutions (dissipative or measure-valued solutions). The concept of a weak-strong uniqueness principle in the class of generalized solutions is used to prove the convergence of various numerical methods. The problem of oscillatory solutions is solved by an original adaptation of the method of K-convergence. An effective method of computing the Young measures is presented. Theoretical results are illustrated by a series of numerical experiments.

Applications of these concepts are to be expected in other problems of fluid mechanics and related fields.

Autorentext

Eduard Feireisl is a senior research worker at the Institute of Mathematics of the Czech Academy of Sciences and full professor at Charles University in Prague. He authored or coauthored 7 research monographs and over 300 research papers registered by the database MathSciNet. His main research interest is the abstract theory of partial differential equations with application in fluid mechanics, including numerical analysis and the effect of stochastic phenomena. He is one of the leading experts in the field of mathematical fluid mechanics.Mária Lukácová-Medvidová is professor for Applied Mathematics at the Johannes Gutenberg-University Mainz. She is a vice speaker of the Research Center Multiscale Simulation Methods for Soft Matter Systems and of the Mainz Institute for Multiscale Modeling. Her research interests lay in numerics and analysis of partial differential equations. She made important contributions to the development of structure-preserving schemes for hyperbolic conservation laws and hybrid multiscale methods for complex fluids. She received various awards, such as the Prize of the Czech Learned Society in 2002, Bronze Medal of the University of KoSice in 2013, or the Gutenberg Research College Fellowship in 2020.

Hana Mizerová is an assistant professor at the Department of Mathematical Analysis and Numerical Mathematics of Comenius University in Bratislava, Slovakia. Her research focuses on numerical analysis of partial differential equations in fluid mechanics. In 2018, she was awarded the Seal of Excellence by the European Commission.

Bangwei She is a research worker at the Institute of Mathematics of the Czech Academy of Sciences. His research interests are in numerical analysis and scientific computing, particularly for partial differential equations with a focus on computational fluid dynamics.

Inhalt

Part I Mathematics of compressible fluid flow: The state-of-the-art.- 1 Equations governing fluids in motion.- 2 Inviscid fluids: Euler system.- 3 Viscous fluids: Navier-Stokes-(Fourier) system.- Part II Generalized solutions to equations and systems describing compressible fluids.- 4 Classical and weak solutions, relative energy.- 5 Generalized weak solutions.- 6 Weak-strong uniqueness principle.- Part III Numerical analysis.- 7 Weak and strong convergence.- 8 Numerical methods.- 9 Finite volume method for the barotropic Euler system.- 10 Finite volume method for the complete Euler system.- 11 Finite volume method for the Navier-Stokes system.- 12 Finite volume method for the barotropic Euler system - revisited.- 13 Mixed finite volume - finite element method for the Navier-Stokes system.- 14 Finite difference method for the Navier-Stokes system.