Efficient High-Order Discretizations for Computational Fluid Dynamics (CISM International Centre for Mechanical Sciences, 602)

✍ Scribed by Martin Kronbichler (editor), Per-Olof Persson (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 314
• Category: Library

✦ Synopsis

The book introduces modern high-order methods for computational fluid dynamics. As compared to low order finite volumes predominant in today's production codes, higher order discretizations significantly reduce dispersion errors, the main source of error in long-time simulations of flow at higher Reynolds numbers. A major goal of this book is to teach the basics of the discontinuous Galerkin (DG) method in terms of its finite volume and finite element ingredients. It also discusses the computational efficiency of high-order methods versus state-of-the-art low order methods in the finite difference context, given that accuracy requirements in engineering are often not overly strict.

The book mainly addresses researchers and doctoral students in engineering, applied mathematics, physics and high-performance computing with a strong interest in the interdisciplinary aspects of computational fluid dynamics. It is also well-suited for practicing computational engineers who would like to gain an overview of discontinuous Galerkin methods, modern algorithmic realizations, and high-performance implementations.

✦ Table of Contents

Preface
Contents
1 The Discontinuous Galerkin Method: Derivation and Properties
Introduction
The Main Concepts
Triangulation of the Computational Domain
Mapping to Curved Elements
Polynomial Approximation
Derivation of the DG Weak Form
Connecting Neighboring Elements: Numerical Fluxes
Weak Form and Fluxes for Systems of Equations
A Strong DG Form
Boundary Conditions
Discrete-in-Space/Continuous-in-Time System
Time Integration
Computation of Integrals
Stability and Convergence
Stability by the Energy Method
Theoretical Convergence Results
Convergence on an Analytical Test Case
Convergence on Deformed Mesh
Refining the Mesh or Increasing the Polynomial Degree?
DG Discretizations for Second Derivatives
Convergence Test with Manufactured Solution
Concluding Remarks
References
2 High-Performance Implementation of Discontinuous Galerkin Methods with Application in Fluid Flow
Introduction
Discontinuous Galerkin Algorithms
Application Efficiency
Computational Characterization of DG Schemes
Background on Computer Architecture
From Application Code to Machine Instructions
The Memory Wall
Parallelization of DG Algorithms
Identifying the Performance Limit
What to Measure
Fast Computation of Integrals with Sum Factorization
Naive Interpolation and Differentiation
Utilizing the Tensor Product
Operation Counts and Measured Throughput with Sum Factorization
Scaling Within a Compute Node
Roofline performance evaluation
Fast Inversion of DG Mass Matrices on Hexahedral Elements
Massively Parallel Computations
Trends and Perspectives
The Euler Equations
A Modern C++ Implementation—The Deal.II Step-67 Tutorial Program
Acoustic Wave Equation
Computational Challenges for Wave Propagation
Solving Linear Systems
Preconditioners
Multigrid Solvers and Preconditioners
Research Trends
The Incompressible Navier–Stokes Equations
Time Integration
Discretization in Space
Stability
Pressure Robustness and H(div) Conforming Schemes
Computational Examples
Perspectives
References
3 Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes Equations
Prologue
Nomenclature
Spectral Calculus Toolbox
Legendre Polynomials and Series
Legendre Polynomial Interpolation
Legendre Gauss Quadrature and the Discrete Inner Product
Aliasing Error
Spectral Differentiation
Spectral Accuracy
The Discrete Inner Product and Summation-by-Parts
Extension to Multiple Space Dimensions
Summary
The Compressible Navier–Stokes Equations
Boundedness of Energy and Entropy
Construction of Curvilinear Spectral Elements
Subdividing the Domain: Spectral Element Mesh Generation
Mapping Elements from the Reference Element
Transforming Equations from Physical to Reference Domains
Building a Modern Discontinuous Galerkin Spectral Element Approximation
Role of the Split Form Approximation
The Importance of the Metric Identities
The Concept of Flux Differencing and Two-Point Fluxes
The Final Assembly: A Robust DGSEM
The Choice of the Two-Point Flux
The Boundedness of the Discrete Entropy
Epilogue
References
4 p-Multigrid High-Order Discontinuous Galerkin Solution of Compressible Flows
Introduction
Compressible RANS Equations
Discontinuous Galerkin Approximation of the RANS and k-widetildeω Turbulence Model Equations
Space Discretization
Computation of the Steady-State Solution
p-Multigrid Solution Strategy
Solution and Error Transfer Operators
Residual and Matrix Restriction Operator
Smoothers
Line Creation Algorithm
Fourier Analysis of the p-multigrid Scheme
Results—Laminar Test Cases
Laminar Delta Wing
Streamlined 3-D Body
Results—Turbulent Test Cases
Turbulent Delta Wing
Train Head
T106A
Conclusion
References
5 High-Order Accurate Time Integration and Efficient Implicit Solvers
Introduction
High-Order Time-Stepping
Semi-Discrete Formulation
Explicit Methods
Backward-Differentiation Formulas (BDF)
(Diagonally) Implicit Runge–Kutta Methods
Implicit Solvers
Jacobian Matrices
Incomplete LU Preconditioning
Minimum Discarded Fill Element Ordering
Performance
Parallelization
Implicit–Explicit (IMEX) Time-Integration
Implicit–Explicit Runge–Kutta Methods
Mesh-Size-Based Splitting of Residual
Quasi-Newton and Preconditioned Krylov Methods
Numerical Results
References
6 An Introduction to the Hybridizable Discontinuous Galerkin Method
Introduction
Laplace Equation
Convergence and Postprocess for Superconvergent Approximation u*
Sparsity Pattern of the HDG Matrix and Computational Efficiency
Incompressible Flow
Matrix Structure and Computational Efficiency
Some Comments on Navier–Stokes
References
7 High-Order Methods for Simulations in Engineering
Introduction
Computation in the Engineering Context
The Life Cycle of Products
Information and Calculation
Mathematical Models of Nature and Their Uncertainty/Errors
Basic Model Uncertainty/Errors: Physics
Basic Model Uncertainty/Errors: Numerical Analysis
Model Parameter Uncertainty/Errors
Model Boundary Condition Uncertainty/Errors
Model Geometry Uncertainty/Errors
Objections to High-Order Methods
Objection 1: Monotonicity
Objection 2: Stencil Size and Shape
Objection 3: Real Order of Accuracy for Nonlinear Cases
Objection 4: Accuracy When Butterfly Effects or Rogue Loads Are Present
Objection 5: Cost Versus Accuracy
Work Estimates for High-Order Schemes
Basic Assumptions
Relative Error
Work Estimates
Possible Objections
Discussion
LES Observations
Taylor Green Vortex
Conclusions
References
Index