Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 Selected Papers from the ICOSAHOM Conference, London, UK, July 9-13, 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide...

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Bibliographic Details
Main Authors: Sherwin, Spencer J, Moxey, David, Peiró, Joaquim, Vincent, Peter E, Schwab, Christoph
Format: eBook
Language:English
Published: Cham Springer Nature 2020
Springer International Publishing AG
Edition:1
Series:Open Access E-Books
Subjects:
Mathematics
Mathematics. Analysis
Nonfiction
ISBN:3030396479, 9783030396473, 3030396460, 9783030396466
Online Access:Get full text
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Table of Contents:
  • 3.3 Spectral Symbol of Curl-Div Matrices Ap,α,βn,G -- 4 Numerical Example -- 5 Conclusions -- References -- Performance of Preconditioners for Large-Scale Simulations Using Nek5000 -- 1 Introduction -- 2 Problem Description -- 3 Optimal Parameter Selection -- 4 Scaling Results -- 5 Conclusions -- References -- Two Decades Old Entropy Stable Method for the Euler Equations Revisited -- 1 Introduction, Objectives and Preliminaries -- 2 Entropy Splitting of the Euler Flux Derivatives -- 3 Semi-Discrete Entropy Split Discretization of the Euler Equations -- 4 Numerical Experiments -- References -- A Mimetic Spectral Element Method for Free Surface Flows -- 1 Introduction -- 1.1 Boundary Conditions -- 2 Discretization of Governing Equations -- 2.1 Basis Functions -- 2.2 Mimetic Discretization in 2D -- 3 Numerical Results -- 4 Discussion and Conclusion -- References -- Spectral/hp Methodology Study for iLES-SVV on an Ahmed Body -- 1 Introduction -- 2 Objectives -- 3 Spectral/hp iLES-SVV -- 4 Simulation Methodology -- 5 Results -- 5.1 Drag Coefficient Comparison Results -- 5.2 Lift Coefficient Comparison Results -- 5.3 Flow Structure Comparison -- 6 Conclusions -- References -- A High-Order Discontinuous Galerkin Solver for Multiphase Flows -- 1 Introduction -- 2 Governing Equations -- 3 Numerical Methods -- 3.1 Spatial Discretisation Using a Nodal Discontinuous Galerkin Scheme (DGSEM) -- 3.2 Time Integration Using IMplicit-EXplicit (IMEX) and Runge-Kutta Schemes -- 4 Validation -- 4.1 Spinodal Decomposition -- 4.2 Rising Bubble -- 5 Conclusions -- References -- High-Order Propagation of Jet Noise on a Tetrahedral Mesh Using Large Eddy Simulation Sources -- 1 Introduction -- 2 Numerical Methods and Solvers -- 2.1 APE Solver -- 2.2 LES Solver -- 2.3 Coupling of Solvers -- 3 Test Cases -- 3.1 Spinning Vortex Pair -- 3.2 3D Turbulent Isolated Jet Noise
  • Intro -- Preface -- Contents -- Part I Invited Papers -- Stability of Wall Boundary Condition Procedures for Discontinuous Galerkin Spectral Element Approximations of the Compressible Euler Equations -- 1 Introduction -- 2 The Compressible Euler Equations and the Wall Boundary Condition -- 3 Stability Bounds for the DGSEM -- 3.1 Linear Stability of Wall Boundary Condition Approximations -- 3.2 Entropy Stability of Wall Boundary Condition Approximations -- 3.2.1 Exact Solution of the Riemann Problem -- 3.2.2 Using Approximate Riemann Solvers for the Boundary Flux -- 4 Discussion -- References -- On the Order Reduction of Entropy Stable DGSEM for the Compressible Euler Equations -- 1 Introduction -- 2 Numerical Model -- 3 Convergence Results -- 3.1 Standard DGSEM -- 3.2 Entropy Conservative and Entropy Stable DGSEM -- 4 Conclusions -- Appendix -- Additional Convergence Results -- Manufactured Solution with Source Term -- References -- A Review of Regular Decompositions of Vector Fields: Continuous, Discrete, and Structure-Preserving -- 1 Introduction -- 2 Preliminaries -- 3 Regular Decompositions and Potentials Related to H(curl) -- 3.1 Gradient-Based Regular Decomposition of H(curl) -- 3.2 Regular Potentials for Some Divergence-Free Functions -- 3.3 Rotation-Bounded Regular Decomposition of H(curl) -- 4 Regular Decompositions and Potentials Related to H(div) -- 4.1 Rotation-Based Regular Decomposition of H(div) -- 4.2 Regular Potential with Prescribed Divergence -- 4.3 Divergence-Bounded Regular Decompositions of H(div) -- 5 Discrete Counterparts of the Regular Decompositions -- 5.1 Discrete Regular Decompositions for Edge Elements -- 5.2 Discrete Regular Decompositions for Face Elements -- References -- Model Reduction by Separation of Variables: A Comparison Between Hierarchical Model Reduction and Proper Generalized Decomposition
  • 3.2 Enforcing Continuity in the Explicit Stage -- 4 Boundary Conditions -- References -- On the Use of Hermite Functions for the Vlasov-Poisson System -- 1 Introduction -- 2 The Continuous Model -- 3 Phase-Space Discretization -- 4 Discretization of the Vlasov Equation -- 5 Numerical Experiments -- References -- HPS Accelerated Spectral Solvers for Time Dependent Problems: Part I, Algorithms -- 1 Introduction -- 2 Time Dependent Boundary Conditions -- 3 Schrödinger Equation -- 4 Burgers' Equation in Two Dimensions -- 5 Conclusion -- References -- High-Order Finite Element Methods for Interface Problems: Theory and Implementations -- 1 Introduction -- 2 XFE and DG Schemes for Interface Problems -- 3 An Optimal Multigrid Method for (7) -- 4 Numerical Tests -- 4.1 High-Order Numerical Quadratures on ``cut'' Elements -- 4.2 2-D Numerical Examples -- 4.3 3-D Numerical Examples -- References -- Stabilised Hybrid Discontinuous Galerkin Methods for the Stokes Problem with Non-standard Boundary Conditions -- 1 Introduction -- 1.1 Notations and Model Problem -- 2 The Stabilised Method -- 2.1 Well-Posedness of the Discrete Problem -- 2.2 Error Analysis -- 3 Numerical Experiments -- 4 Conclusion -- References -- RBF Based CWENO Method -- 1 Introduction -- 2 CWENO -- 3 Radial Basis Functions -- 4 RBF-CWENO -- 4.1 Smoothness Indicator -- 5 Numerical Results -- 5.1 Linear Advection Equation -- 5.2 Burger's Equation -- 5.3 Euler Equations -- 5.3.1 Sod's Shock Tube Problem -- 5.3.2 Shu-Osher Shock-Entropy Wave Interaction Problem -- 6 Conclusion -- References -- Discrete Equivalence of Adjoint Neumann-Dirichlet div-grad and grad-div Equations in Curvilinear 3D Domains -- 1 Introduction -- 2 Function Spaces -- 2.1 Primal Polynomial Spaces -- 2.2 Algebraic Dual Polynomial Spaces -- 2.3 Function Spaces in Curvilinear Domains -- 3 Weak Formulations
  • 3.1 Discrete Neumann Problem -- 3.2 Discrete Dirichlet Problem -- 3.3 Equivalence Between Discrete Formulations -- 4 Numerical Test -- 5 Conclusions -- References -- A Conservative Hybrid Method for Darcy Flow -- 1 Introduction -- 2 Darcy Flow Formulation -- 2.1 Notations -- 2.2 Weak Formulation -- 3 Basis Functions -- 3.1 Primal and Dual Nodal Degrees of Freedom -- 3.2 Primal and Dual Edge Degrees of Freedom -- 3.3 Differentiation of Nodal Polynomial Representation -- 4 Discrete Inner Product and Duality Pairing -- 4.1 Weighted Inner Product -- 4.2 Divergence of Velocity -- 4.3 Connectivity Matrix -- 5 Discrete Formulation -- 6 Results -- References -- High-Order Mesh Generation Based on Optimal Affine Combinations of Nodal Positions -- 1 Introduction -- 2 High-Order Mesh Generation Based on Affine Combinations of Nodal Positions -- 3 Numerical Experiments -- 4 Concluding Remarks and Future Work -- References -- Sparse Spectral-Element Methods for the Helically Reduced Einstein Equations -- 1 Introduction -- 2 Background -- 2.1 Helically Reduced Wave Equation -- 2.2 Helically Reduced Einstein Equations -- 3 Sparse Spectral Element Methods -- 3.1 Overview -- 3.2 Integration Preconditioning and Other Key Aspects -- 3.3 Current Complexity Estimates -- 4 Numerical Tests -- 4.1 Comparison with Exact Solutions -- 4.2 Gauge Constraint Tests -- 5 Conclusions and Acknowledgments -- Appendix: Beetle, Bromley, Hernández, and Price Inner Boundary Conditions -- References -- Spectral Analysis of Isogeometric Discretizations of 2D Curl-Div Problems with General Geometry -- 1 Introduction -- 2 Preliminaries -- 2.1 Spectral Distribution -- 2.2 B-Splines -- 3 Spectral Analysis of Isogeometric Discretizations in 2D -- 3.1 Matrices Related to a Generalized Poisson Problem -- 3.2 Matrices Related to Our Curl-Div Problem
  • 4 Conclusions
  • 1 Introduction -- 2 The HiMod Approach -- 3 The PGD Approach -- 4 HiMod Reduction Versus PGD -- 4.1 Domain Geometry -- 4.2 Modeling of the Transverse Dynamics -- 4.3 Computational Aspects -- 5 HiMod Reduction and PGD for Parametrized Problems -- References -- Recurrence Relations for a Family of Orthogonal Polynomials on a Triangle -- 1 Introduction -- 2 Ladder Operators for Jacobi Polynomials -- 3 Ladder Operators for Pn,k(a,b,c,d) -- 4 Sparse Recurrence Relations for Pn,k(a,b,c) -- 4.1 Differentiation -- 4.2 Conversion -- 4.3 Multiplication -- 4.4 Differential Eigenvalue Problems -- 5 Application: Calculating Laplacians -- 6 Conclusion -- References -- Part II Contributed Papers -- Greedy Kernel Methods for Center Manifold Approximation -- 1 Introduction -- 2 Background -- 3 Kernel Approximation -- 3.1 Greedy Approximation -- 4 Numerical Examples -- 4.1 Example 1 -- 4.2 Example 2 -- 4.3 Example 3 -- 5 Conclusions -- References -- An Adaptive Error Inhibiting Block One-Step Method for Ordinary Differential Equations -- 1 Introduction -- 2 Error Inhibiting Block One-Step Method -- 3 RBF Interpolation -- 4 Construction of the Adaptive Error Inhibiting Scheme -- 5 Numerical Results -- 6 Conclusions -- References -- Hermite Methods in Time -- 1 Introduction -- 2 Basic Properties of Hermite-Birkhoff Interpolation -- 3 Time-Stepping Methods -- 3.1 Stability and Consistency -- 4 Numerical Experiments -- 4.1 Arentsorf Orbit -- 4.2 Van der Pol Oscillator -- 4.3 Schrödinger Equation -- 5 Conclusions and Future Work -- References -- HPS Accelerated Spectral Solvers for Time Dependent Problems: Part II, Numerical Experiments -- 1 Introduction -- 2 The Hierarchical Poincaré-Steklov Method -- 2.1 Discretization -- 2.2 A Hierarchical Direct Solver -- 3 Time-Stepping Methods -- 3.1 Neumann Data Correction in the Slope Formulation