Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems

Description

A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021. Preface Contents Contributors A Tree Structure Approach to Reachability Analysis 1 Introduction 2 Problem Setup 3 Backwards Reachability 4 Forwards Reachability 5 A Tree-Based Algorithm for Computing Reachable Sets 6 Numerical Examples 6.1 Numerical Example 1: Linear System with Two States 6.2 Numerical Example 2: DC Motor 7 Conclusions and Future Works References Asymptotic-Preserving Neural Networks for Hyperbolic Systems with Diffusive Scaling 1 Introduction 2 Hyperbolic Systems with Diffusive Scaling 3 Review of Deep Neural Networks and Physics-Informed Neural Networks 3.1 Deep Neural Networks (DNNs) 3.2 Physics-Informed Neural Networks (PINNs) 4 Asymptotic-Preserving Neural Networks 4.1 APNN for the Goldstein-Taylor Model 5 Application to the Goldstein-Taylor Model 5.1 Standard DNN Versus Standard PINN in Hyperbolic Regime 5.2 Standard PINN Versus APNN in Diffusive Regime 6 Application to Epidemic Dynamics 6.1 The Multiscale Hyperbolic SIR Model 6.2 APNN for the Hyperbolic SIR Model 6.3 APNN Performance with Epidemic Dynamics 7 Conclusion References A Non-local System Modeling Bi-directional Traffic Flows 1 Introduction 2 A Non-local Bi-directional Traffic Flow Model 3 Existence of Weak Solutions 4 Numerical Tests 4.1 Kernel Support Tending to Zero 4.2 Asymptotic Behaviour in a Periodic Setting 4.3 Maximum Principle References Semi-implicit Finite-Difference Methods for Compressible Gas Dynamics with Curved Boundaries: A Ghost-Point Approach 1 Introduction 2 Finite-Difference Methods for Conservation Laws 2.1 Compressible Euler Equations of Gas Dynamics 2.2 Explicit and Semi-implicit Spatial Discretization 2.3 Explicit and Semi-implicit Time Discretization 2.4 Discretization of Compressible Euler Equations in 2D 3 Ghost-Point Method for Boundary Conditions 3.1 Ghost-Point Technique for Implicit Solvers 4 Numerical Simulations 4.1 Square Obstacle 4.2 Circular Obstacle 4.3 Shock Tube Problems 5 Conclusions References High-Order Arbitrary-Lagrangian-Eulerian Schemes on Crazy Moving Voronoi Meshes 1 Introduction 1.1 Goals 1.2 Structure 2 Hyperbolic Partial Differential Equations 3 Numerical Method 3.1 Direct Arbitrary-Lagrangian-Eulerian Schemes 3.2 Topology Changes and Crazy Sliver Elements 3.3 ADER-ALE Algorithm: The Predictor Step 3.4 A Posteriori Sub-cell FV Limiter 4 Numerical Examples 4.1 Long Time Evolution of a Shu-type Vortical Equilibrium 4.2 Sedov Explosion Problem 4.3 Traveling Sod-type Explosion Problem 5 Conclusion and Outlook References Overview on Uncertainty Quantification in Traffic Models via Intrusive Method 1 Introduction 2 Stochastic Galerkin Approach 3 Microscopic Scale 4 Mesoscopic Scale 5 Macroscopic Scale 5.1 From Micro to Macro 5.2 From Meso to Macro 5.3 Numerical Test 6 Conclusion and Future Perspectives References A Study of Multiscale Kinetic Models with Uncertainties 1 Introduction 2 Mathematical Theory for Uncertain Kinetic Equations 2.1 Theoretical Framework: The Perturbative Setting 2.2 Convergence to the Global Equilibrium 3 Stochastic Galerkin Method: An Intrusive Scheme 3.1 Error Analysis of the gPC-SG System 3.2 Stochastic AP Schemes 4 Multi-fidelity Method: A Non-intrusive Scheme 4.1 A Bi-fidelity Stochastic Collocation (BFSC) Algorithm 4.2 Numerical Examples 5 Conclusion References On the Shock Wave Discontinuities in Grad Hierarchy for a Binary Mixture of Inert Gases 1 Introduction 2 13–Moment Equations and Principal Subsystems 3 The Shock Wave Problem 3.1 Singularity Manifolds and Critical Mach Numbers 4 Singularity Analysis References A Conservative a-Posteriori Time-Limiting Procedure in Quinpi Schemes 1 Introduction 2 Quinpi Scheme for Hyperbolic Conservation Laws 2.1 The Quinpi Approach 3 Numerical Tests 3.1 Test 1: Experimental Order of Convergence 3.2 Test 2: Linear Transport Problem 3.3 Test 3: Burgers Equation 3.4 Computational Performance of the Quinpi Schemes 4 Conclusion and Perspectives References Applications of Fokker Planck Equations in Machine Learning Algorithms 1 Introduction 2 Algorithmic Fairness for Imbalanced Data 2.1 Setting 2.2 Theorems 3 Asynchronous Stochastic Gradient Descent 3.1 Setting 3.2 Theorems 4 Reinforcement Learning in Smooth Environment 4.1 Setting 4.2 Theorems 5 Conclusion References