Recent Advances in Numerical Methods for Hyperbolic PDE Systems: NumHyp 2019 (SEMA SIMAI Springer Series, 28)

✍ Scribed by María Luz Muñoz-Ruiz (editor), Carlos Parés (editor), Giovanni Russo (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 272
• Category: Library

✦ Synopsis

The present volume contains selected papers issued from the sixth edition of the International Conference "Numerical methods for hyperbolic problems" that took place in 2019 in Málaga (Spain). NumHyp conferences, which began in 2009, focus on recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations (PDEs) and their applications. The 11 chapters of the book cover several state-of-the-art numerical techniques and applications, including the design of numerical methods with good properties (well-balanced, asymptotic-preserving, high-order accurate, domain invariant preserving, uncertainty quantification, etc.), applications to models issued from different fields (Euler equations of gas dynamics, Navier-Stokes equations, multilayer shallow-water systems, ideal magnetohydrodynamics or fluid models to simulate multiphase flow, sediment transport, turbulent deflagrations, etc.), and the development of new nonlinear dispersive shallow-water models.

The volume is addressed to PhD students and researchers in Applied Mathematics, Fluid Mechanics, or Engineering whose investigation focuses on or uses numerical methods for hyperbolic systems. It may also be a useful tool for practitioners who look for state-of-the-art methods for flow simulation.

✦ Table of Contents

Preface
Contents
Numerical Methods for General Problems
Incomplete Riemann Solvers Based on Functional Approximations to the Absolute Value Function
1 Introduction
2 Approximate Viscosity Matrix (AVM) Methods
3 Approximate DOT Solvers
4 The Nonconservative Case
5 Numerical Experiments
5.1 Applications to Magnetohydrodynamics
5.2 Applications to the Two-Layer Shallow Water System
References
Entropy–Based Methods for Uncertainty Quantification of Hyperbolic Conservation Laws
1 Introduction
2 Why Galerkin-Type Intrusive Methods?
3 Hyperbolic Conservation Laws and the IPM Method
3.1 Hyperbolic Conservation Laws and Entropy Variables
3.2 The Intrusive Polynomial Moment Method
4 Realizability-Preserving Spatial Discretization
4.1 Realizability
4.2 Realizability-Preserving Discretization
5 Accelerating the IPM Solution
5.1 Adaptivity
5.2 One-Shot IPM
6 Results
6.1 Burgers' Equation
6.2 Euler Equations
6.3 2-D Euler Equations with One-Shot
6.4 Unsteady 2-D Euler Equations with Adaptivity
References
Well-Balanced Reconstruction Operator for Systems of Balance Laws: Numerical Implementation
1 Introduction
2 Control Problem
3 Numerical Algorithms
3.1 Newton's Method
3.2 Descent Methods
3.3 Numerical Integration
4 A Numerical Test for the Control Problem
5 Numerical Experiments
5.1 Burgers Equation with a Nonlinear Source Term
5.2 Shallow Water Equations
6 Conclusions
References
On High-Precision Linfty-stable IMEX Schemes for Scalar Hyperbolic Multi-scale Equations
1 Introduction
2 IMEX Formulation
2.1 A 2-Stage CK Type IMEX R-K Method
2.2 A 3-Stage CK Type IMEX R-K Method
3 Numerical Results
3.1 Optimal Order Detection: A MOOD-like Technique
3.2 Choice of Beta in the 3-stage method
3.3 Numerical Tests
4 Conclusions and Future Work
References
Numerical Methods for Specific Problems
A Staggered Pressure Correction Numerical Scheme to Compute a Travelling Reactive Interface in a Partially Premixed Mixture
1 Problem Position
2 The Physical Models
3 General Description of the Scheme and Main Results
4 Meshes and Unknowns
5 The Scheme
5.1 Euler Step
5.2 Chemistry Step
6 Scheme Conservativity
7 Numerical Tests
8 Appendix A: The MUSCL Scheme
9 Appendix B: An Anti-diffusive Scheme
References
New Invariant Domain Preserving Finite Volume Schemes for Compressible Flows
1 Introduction
2 Finite Volume Schemes
2.1 Mesh and Space Discretization
2.2 Numerical Scheme for the Euler System
2.3 Numerical Scheme for the Navier–Stokes–Fourier System
3 Numerical Experiments
3.1 Numerical Experiments for the FLM Method
3.2 Numerical Experiments for the FV Method for NSF
References
Recent Advances and Complex Applications of the Compressible Ghost-Fluid Method
1 Introduction
2 Governing Equations
3 Numerical Method
3.1 Building Blocks for the Level-Set Ghost-Fluid Method
3.2 The Level-Set Ghost-Fluid Method
4 Numerical Results
4.1 Drop Collision in Linear Velocity Profile
4.2 Droplet Collisions with Wec=2.2
4.3 Shock-Droplet Interaction
5 Conclusion
References
Entropy Stable Numerical Fluxes for Compressible Euler Equations Which Are Suitable for All Mach Numbers
1 Introduction
2 Kinetic Energy and Entropy Stable Fluxes
2.1 Entropy-Entropy Flux and Entropy Variables
2.2 A Basic Finite Volume Method
2.3 Entropy Conservative Numerical Fluxes
2.4 Kinetic Energy Preserving Fluxes
2.5 Entropy Diffusion
2.6 Kinetic Energy Diffusion
2.7 Intermediate State
3 Low Mach Number Asymptotics
4 Low Mach Modifications of the ES and ES-KES Fluxes
5 Numerical Tests
5.1 Test of the Entropy Stability
5.2 Test of the Contact Property
5.3 Low Mach Gresho Vortex
6 Conclusions and Outlook
References
Residual Based Method for Sediment Transport
1 Introduction
2 A Sediment Transport in a Shallow Water
3 A Residual Based Predictor-Corrector Upwind Discretization for 1D Space-Time Sediment Transport
4 Numerical Results
4.1 Test of Well-Balanced Property
4.2 A 1D Space-Time Dune Test Case
4.3 A Test of Order
4.4 A Dam Break Test over a Wet Bottom Topography
5 Concluding Remarks
References
New Flow Models
Pseudo-compressibility, Dispersive Model and Acoustic Waves in Shallow Water Flows
1 Presentation
2 A Compressible and Dispersive Model in Shallow Water Context
2.1 The Compressible Navier-Stokes-Fourier System
2.2 Thermodynamic Considerations
2.3 Acoustic Waves and Water Waves
2.4 Sound Speed for Sea Water
2.5 A Shallow Water Approximation of the Compressible Euler System
2.6 When the Density is Almost Constant
2.7 The Boundary Conditions
2.8 Dispersion Relation
2.9 A Pseudo-compressible Model
3 The Numerical Scheme (Explicit in Time)
3.1 Time Discretisation
3.2 Influence of the Pseudo-compressibility over the Computational Costs
3.3 Choice of ε
4 Detailed Numerical Scheme in 1d
4.1 Semi-discrete (in Time) Scheme
4.2 The Semi-discrete (in Space) Scheme
4.3 Wet-Dry Interface
4.4 Stability of the Scheme
4.5 Simulation Results
References
A Generalised Serre-Green-Naghdi Equations for Variable Rectangular Open Channel Hydraulics and Its Finite Volume Approximation
1 Introduction
2 The Three-Dimensional Incompressible Euler Equations
2.1 Settings
2.2 Dimensionless Euler Equations
3 Derivation of the Section-Averaged Model
3.1 Width-Averaged Equations
3.2 Depth-Averaged Equations
3.3 Section-Averaged Model
4 A Well-Balanced Finite Volume Approximation
4.1 Numerical Method
4.2 Propagation of a Solitary Wave
5 Conclusions and Perspectives
References
Author Index