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Full Description
The following chapters summarize lectures given in March 2001 during the summerschool on Hyperbolic Partial Differential Equations which took place at the Technical University of Hamburg-Harburg in Germany. This type of meeting is originally funded by the Volkswa genstiftung in Hannover (Germany) with the aim to bring together well-known leading experts from special mathematical, physical and engineering fields of interest with PhD students, members of Scientific Research Institutes as well as people from Industry, in order to learn and discuss modern theoretical and numerical developments. Hyperbolic partial differential equations play an important role in various applications from natural sciences and engineering. Starting from the classical Euler equations in fluid dynamics, several other hyperbolic equations arise in traffic flow problems, acoustics, radiation transfer, crystal growth etc. The main interest is concerned with nonlinear hyperbolic problems and the special structures, which are characteristic for solutions of these equations, like shock and rarefaction waves as well as entropy solutions. As a consequence, even numerical schemes for hyperbolic equations differ significantly from methods for elliptic and parabolic equations: the transport of information runs along the characteristic curves of a hyperbolic equation and consequently the direction of transport is of constitutive importance. This property leads to the construction of upwind schemes and the theory of Riemann solvers. Both concepts are combined with explicit or implicit time stepping techniques whereby the chosen order of accuracy usually depends on the expected dynamic of the underlying solution.
Contents
1 Hyperbolic Conservation Laws and Industrial Applications.- 1.1 Transport theorem and balance laws.- 1.2 Linear initial and boundary value problems.- 1.3 Weak solutions and entropy.- 1.4 Systems of conservation laws.- 2 Central Schemes and Systems of Balance Laws.- 2.1 Second order central schemes.- 2.2 High order central schemes.- 2.3 Multidimensional central schemes.- 2.4 Treatment of the source.- Further developments.- 3 Methods on unstructured grids, WENO and ENO Recovery techniques.- 3.1 Introduction to finite volume approximations.- 3.2 Governing equations.- 3.3 Finite volume approximations.- 3.4 Time stepping schemes.- 3.5 Remarks on the philosophy of ENO schemes.- 3.6 Polynomial recovery.- 3.7 WENO approximations.- 3.8 The theory of optimal recovery.- 3.9 Grid adaptivity for box methods.- 3.10 Error and residual.- 3.11 Experience with L2.- 3.12 The dual graph-norm.- 3.13 Closing of the circle: L2 meets dual graph norm.- 4 Pressure-Correction Methods for all Flow Speeds.- 4.1 Introduction.- 4.2 Conservation Equations.- 4.3 Pressure-Correction Equation for Incompressible Flows.- 4.4 Pressure-Correction Equation for Compressible Flows.- 4.5 Solution Algorithm for all Flow Speeds.- 4.6 FV-Method for Arbitrary Control Volumes.- 4.7 Pressure-Correction Algorithm for FV-Methods.- 4.8 Implementation of Boundary Conditions.- 4.9 Examples of Application.- 4.10 Conclusions.- 5 Computational Fluid Dynamics and Aeroacoustics for Low Mach Number Flow.- 5.1 Introduction.- 5.2 Non-Dimensionalisation of the Governing Equations.- 5.3 The Incompressible Limit of a Compressible Fluid Flow.- 5.4 Numerical Methods for Low Mach Number Fluid Flow.- 5.5 Sound Generation and Sound Propagation.- 5.6 Multiple Scale Considerations.- 5.7 Numerical Aeroacoustics.- 5.8 Conclusions.
