Adaptive High-Order Methods in Computational Fluid Dynamics 7792tp.indd 2/9/11 3:15 PM Advances in Computational Fluid Dynamics Editors-in-Chief: Chi-Wang Shu (Brown University, USA) and Chang Shu (National University of Singapore, Singapore) Published Vol Adaptive High-Order Methods in Computational Fluid Dynamics edited by Z J Wang (Iowa State University, USA) Forthcoming Vol Computational Methods for Two-Phase Flows by Peter D M Spelt (Imperial College London, UK), Stephen J Shaw (X'ian Jiaotong – University of Liverpool, Suzhou, China) & Hang Ding (University of California, Santa Barbara, USA) Steven - Adaptive High-Order Methods.pmd 2/1/2011, 11:47 AM Vol Advances in Computational Fluid Dynamics Adaptive High-Order Methods in Computational Fluid Dynamics Editor Z J Wang Iowa State University, USA World Scientific NEW JERSEY 7792tp.indd • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 2/9/11 3:15 PM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ADAPTIVE HIGH-ORDER METHODS IN COMPUTATIONAL FLUID DYNAMICS Advances in Computational Fluid Dynamics — Vol Copyright © 2011 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN-13 978-981-4313-18-6 ISBN-10 981-4313-18-1 Printed in Singapore Steven - Adaptive High-Order Methods.pmd 2/1/2011, 11:47 AM To My Family This page intentionally left blank Preface This book contains invited chapters written by leading international experts on adaptive high-order methods in computational fluid dynamics (CFD) It covers several widely used, and still intensively researched methods, including the discontinuous Galerkin (DG), residual distribution, differential quadrature, k-exact finite volume, spectral volume/spectral difference, PNPM, and correction procedure via reconstruction methods The reasons for including such a wide coverage of methods are to: (1) provide a single source of reference, (2) present a snapshot of the state-of-the-art, and (3) facilitate the observation of similarities and differences as well as pros and cons of these methods In the present context, adaptive high-order methods refer to numerical methods that are capable of handling unstructured adaptive meshes with accuracy higher than second-order These methods are compact, scalable, capable of handling both complex physics and geometry, and suitable for modern parallel supercomputers and graphics processing units (GPUs) They are widely considered the next major breakthrough in CFD, and have already found applications in computational aeroacoustics, computational electromagnetics, vortex dominated flows, and large eddy simulation and direct numerical simulation of turbulent flows A concerted effort was made to minimize overlaps among the chapters For example, the first chapters describe different aspects of the DG methods, while the last chapters are devoted to other high-order methods Main topics covered include innovative formulations, analyses, efficient solution and time marching algorithms, parallel implementation, turbulence modeling, discontinuity-capturing techniques, error estimates, hp-adaptations, and dynamic mesh techniques, etc The book requires a graduate student level of understanding It should serve as an excellent source of information for CFD developers, educators, researchers, users, and students who are interested in the stateof-the-art and the remaining challenges in adaptive high-order methods vii viii Preface I am grateful to Dr Chang Shu, a close friend and the Co-Editor-inChief of the book series, Advances in Computational Fluid Dynamics in World Scientific, for suggesting the book Heartfelt thanks are due to all the contributors of this volume Needless to say, the book would not exist without their hard work Finally, I’d like to thank Ying Zhou for producing the color cover graphic, and Varun Vikas for help with Latex Z.J Wang Ames, Iowa June 30, 2011 CONTENTS Preface vii Chapter 1: Discontinuous Galerkin for Turbulent Flows Francesco Bassi, Lorenzo Botti, Alessandro Colombo, Antonio Ghidoni And Stefano Rebay Chapter 2: Massively Parallel Solution Techniques for Higher-Order Finite-Element Discretizations in CFD Laslo T Diosady and David L Darmofal 33 Chapter 3: Error Estimation and hp–Adaptive Mesh Refinement for Discontinuous Galerkin Methods Tobias Leicht and Ralf Hartmann 67 Chapter 4: A Runge-Kutta based Discontinuous Galerkin Method with Time Accurate Local Time Stepping Gregor J Gassner, Florian Hindenlang and Claus-Dieter Munz 95 Chapter 5: High-Order Discontinuous Galerkin Methods for CFD Jaime Peraire and Per-Olof Persson 119 Chapter 6: Weighted Non-Oscillatory Limiters for Runge-Kutta Discontinuous Galerkin Methods Jianxian Qiu 153 Chapter 7: A Venerable Family of Discontinuous Galerkin Schemes for Diffusion Revisited Bram van Leer, Marcus Lo, Rita Gitik and Shohei Nomura 185 ix November 23, 2010 446 13:50 World Scientific Review Volume - 9in x 6in Z J Wang, H Gao & T Haga The laminar boundary layer over a plate is computed on a prism mesh The Reynolds number based on the plate length L is ReL = 10, 000 and the freestream Mach number is M = 0.2 The boundary layer thickness at the √ trailing edge is estimated by the approximate relation δ = 5L/ ReL The computational domain is selected to be (−2 ≤ x ≤ 1, ≤ y ≤ 100δ, ≤ z ≤ δ) , with L = Note that the domain size in the y-direction is chosen to be large enough to not significantly affect the computational results especially in the v-velocity profiles The prism mesh was produced from a Cartesian grid, with clustering at the wall and near the leading edge In the spanwise z-direction, only one cell was generated Figure 11(a) shows the computed Mach number using polynomials of degree in the y-direction and polynomials of degree in x- and z- directions The grid has only two cells in the boundary layer at x = 1.0 and 17 cells along the plate The numbers of prism cells and DOFs are 728 and 26208 respectively In comparison, a finer grid was generated by dividing each prism cell into two prism cells to have twice the number of cells in the y- direction We employed degree polynomials in all directions on this finer grid Since each prism cell of degree polynomials has solution points, which is half of the solution points for the degree polynomials, the total number of DOFs is the same as the case using degree polynomials in the normal direction The computed Mach number on this finer grid is shown in Fig 11(b) In Fig 12, the computed v-velocity profiles in the boundary layer at x = and skin friction profiles along the plate are shown As we expected, (a) k = in the y-direction (b) k = on a finer mesh in the y-direction Fig 11 Mach number contours of a laminar boundary layer on a flat plate (enlarged by a factor of 10 in y direction) 15˙chapter-15 November 23, 2010 13:50 World Scientific Review Volume - 9in x 6in 15˙chapter-15 447 A Unifying Discontinuous Formulation for Hybrid Meshes 10-2 Blasius solution 3rd and 6th order hybrid 3rd order 10-2 cf 10-2 10-3 10-3 10-3 10-3 10-3 -0.2 Blasius solution 3rd and 6th order hybrid 3rd order 10-2 0.2 0.4 0.6 0.8 1.2 v(2Rex)1/2/U (a) v velocity profile at x = 0.5 1.4 10-3 0.2 0.4 0.6 0.8 x (b) Skin friction profile Fig 12 Comparison of v-velocity and cf profiles for the flat plate boundary layer problem the computed profiles using the higher order scheme agree better with the Blasius’s solution The convergence histories are compared in Fig 13 The computations were performed using the block preconditioned LU-SGS scheme with several different time steps It is shown that, a larger time step can be taken in the case employing the higher order elements with less grid cells and it takes fewer number of iterations to converge to the steady state 5.4 Unsteady subsonic flow over a sphere at Re=300 Next, we consider an unsteady flow case over a sphere with a Reynolds number of 300 based on the diameter of the sphere The inflow Mach number is 0.3 The hybrid prismatic and tetrahedral computational mesh is shown in Fig 14 To resolve shedding vortices, finer elements are generated in the wake region The total number of cells is 54,312 The local grid size around the sphere is ∼ 0.2r and the size in the wake region is ∼ 0.8r with r the radius of the sphere The computed Q isosurface colored by local Mach number using the 4th-order CPR scheme is shown in Fig 15 The obtained plain symmetric wake vortex structure is comparable to the available experimental and computational results in Refs 14 and 24 at least qualitatively In Fig 16 we plot the history of the drag coefficient Cd in terms of non-dimensional time t The computed drag coefficient and the oscillating amplitude of drag and November 23, 2010 13:50 World Scientific Review Volume - 9in x 6in 448 Z J Wang, H Gao & T Haga 100 3rd-6th t=0.2 3rd-6th t=0.4 3rd-6th t=0.8 3rd-6th t=1.6 3rd t=0.2 3rd t=0.4 3rd t=0.6 10-2 Residual 10-4 10-6 10-8 10-10 10-12 10-14 100 200 300 400 500 Time step Fig 13 lem Comparison of the convergence histories for the flat plate boundary layer prob- (a) Entire grid Fig 14 300 (b) Grid around the sphere Computational grid around a sphere for the unsteady viscous flow at Re = the Strouhal number St are shown in Table For comparison, results from Gassner14 using the 4th-order DG scheme on a tetrahedral grid and from Tomboulides36 and Johnson and Patel24 obtained with an incompressible simulation, are shown as well The results computed with the CPR method agree reasonably well with those reference values 15˙chapter-15 January 25, 2011 15:9 World Scientific Review Volume - 9in x 6in 15˙chapter-15 A Unifying Discontinuous Formulation for Hybrid Meshes 449 Fig 15 Computed Q isosurfaces in the wake region of the viscous laminar flow over a sphere at Re=300 0.68 0.675 0.67 CD 0.665 0.66 0.655 0.65 0.645 0.64 200 400 600 800 1000 1200 1400 Time Fig 16 300 Time history of the drag coefficient for unsteady flow over a sphere at Re = Conclusions This chapter describes a discontinuous method named correction procedure via reconstruction or CPR for hybrid meshes The CPR formulation unifies the discontinuous Galerkin, staggered grid, spectral volume and spectral January 25, 2011 15:9 World Scientific Review Volume - 9in x 6in 450 Z J Wang, H Gao & T Haga Table Comparisons of the averaged drag coefficient, the amplitude of drag and the Strouhal number Method Cd ∆Cd St Present Gassner14 Tomboulides36 Johnson & Patel24 0.670 0.673 0.671 0.656 0.0032 0.0031 0.0028 0.0035 0.131 0.135 0.136 0.137 difference methods into a single differential formulation, and is particular simple for high-order elements The extensions to viscous flow, and to 3D mixed grids are also presented Various accuracy studies have verified the CPR method is capable of obtaining the designed order of accuracy for both inviscid and viscous flow problems Other benchmark and test cases have demonstrated the capability of the method Future work includes the development of efficient, and low memory solvers, and solution based hp-adaptations Acknowledgments The research on high-order methods has been funded by AFOSR grant FA9550-06-1-0146, and partially by DOE grant DE-FG02-05ER25677 References D N Arnold, F Brezzi, B Cockburn and L D Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal 19 (4), pp 742–760, (2002) T J Barth and P O Frederickson, High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA-90-0013, (1990) F Bassi and S Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J Comput Phys 131, pp 267–279, (1997) F Bassi and S Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J Comput Phys 138 (2), pp 251–285, (1997) F Bassi and S Rebay, GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, In eds B Cockburn, G.E Karniadakis and C W Shu, Discontinuous Galerkin methods: Theory, Computations and Applications pp 197-208, Springer, Berlin, (2000) 15˙chapter-15 January 25, 2011 15:9 World Scientific Review Volume - 9in x 6in A Unifying Discontinuous Formulation for Hybrid Meshes 15˙chapter-15 451 R F Chen and Z J Wang, Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids, AIAA J 38 (12), pp 2238–2245, (2000) B Cockburn and C W Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math Comput 52, pp 411–435, (1989) B Cockburn and C W Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J Comput Phys 141, pp 199–224, (1998) B Cockburn and C W Shu, The local discontinuous Galerkin methods for time-dependent convection diffusion systems, SIAM J Numer Anal 35, pp 2440–2463, (1998) 10 M Delanaye and Y Liu, Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids, AIAA-99-3259, (1999) 11 V Dolejˇ´ On the discontinuous Galerkin method for numerical solution of si, the Navier-Stokes equations, Int J Numer Meth Fluids 45, pp 1083–1106, (2004) 12 M Dumbser, PN PM schemes on unstructured meshes for time-dependent partial differential equations In eds Z J Wang, Adaptive High-order Methods in Computational Fluid Dynamics pp 233 World Scientific, Singapore, (2011) 13 J A Ekaterinaris, High-order accurate, low numerical diffusion methods for aerodynamics, Progress in Aerospace Sciences 41, pp 192–300, (2005) 14 G J Gassner, J F Lorcher, C-D Munz and J S Hesthaven, Polymorphic nodal elements and their application in discontinuous Galerkin methods, J Comput Phys 228, pp 1573–1590, (2005) 15 S K Godunov, A finite-difference method for the numerical computation of discontinuos solutions of the equations of fluid dynamics, Math Sbornik 47, pp 271–306, (1959) 16 S Gottlieb and C W Shu, Total variation diminishing Runge-Kutta schemes, Math Comput 67, pp 73–85, (1998) 17 T Haga, H Gao and Z J Wang, A high-order unifying discontinuous formulation for 3D mixed grids, AIAA-2010-540, (2010) 18 R Hartmann and P Houston, Symmetric interior penalty DG emthods for the compressible Navier-Stokes Equations I: Method formulation, Int J Numer Anal Model (1), pp 1–20, (2006) 19 J S Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J Numer Anal 35 (2), pp 655–676, (1998) 20 H T Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA-2007-4079, (2007) 21 H T Huynh, A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion, AIAA-2009-403, (2009) 22 H T Huynh, High-order methods by correction procedures using reconstructions In eds Z J Wang, Adaptive High-order Methods in Computational Fluid Dynamics pp 422 World Scientific, Singapore, (2011) November 23, 2010 452 13:50 World Scientific Review Volume - 9in x 6in Z J Wang, H Gao & T Haga 23 A Jameson, Analysis and design of numerical schemes for gas dynamics I Artificial diffusion, upwind biasing, limiters and their eefect on accuracy and multigrid convergence, Int J Comput Fluid Dyn 4, pp 171–218, (1994) 24 T A Johnson and V C Patel, Flow past a sphere up to a Reynolds number of 300, J Fluid Mech 378, pp 19–70, (1999) 25 G E Karniadakis and S J Sherwin, Spectral-hp Element Methods Oxford University Press, Oxford, England, (1999) 26 D A Kopriva and J H Kolias, A conservative staggered-grid Chebyshev multidomain method for compressible flows, J Comput Phys 125, pp 244– 261, (1996) 27 M.-S Liou, A sequel to AUSM, Part II: AUSM+-up for all speeds, J Comput Phys 214, pp 137–170, (2006) 28 Y Liu, M Vinokur and Z J Wang, Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems, J Comput Phys 212, pp 454–472, (2006) 29 Y Liu, M Vinokur and Z J Wang, Discontinuous spectral difference method for conservation laws on unstructured grids, J Comput Phys., 216, pp 780– 801, (2006) 30 G May and A Jameson, A spectral difference method for the Euler and Navier-Stokes equations, AIAA-2006-304, (1996) 31 A Nejata and C Ollivier-Gooch, A high-order accurate unstrcutured finite voume Newton-Krylov algorithm for inviscid compressible flows, J Comput Phys 227, pp 2582–2609, (2008) 32 J Peraire and P.-O Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J Sci Comput 30, pp 1806–1824, (2008) 33 W H Reed and T R Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report, LA-UR-73-479, (1973) 34 P L Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J Comput Phys 43, pp 357–372, (1981) 35 V V Rusanov, Calculation of interaction of non-steady shock waves with obtsacles, SIAM J Comput Math Phys 1, pp 261–279, (1961) 36 A G Tomboulides and S A Orzag, Numerical investigation of transitional and weak turbulent flow past a sphere, J Fluid Mech 416, pp 45–73, (2000) 37 K Van der Abeele, C Lacor and Z J Wang, On the stability and the accuracy of the spectral difference method, J Sci Comput 37 (2), pp 162– 188, (2008) 38 B Van Leer, Towards the ultimate conservative differences scheme V a second order sequel to Godunov’s method, J Comput Phys 32, pp 101–136, (1979) 39 B Van Leer and S Nomura, Discontinuous Galerkin for diffusion, AIAA2005-5108, (2005) 40 Z J Wang, Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J Comput Phys 178 (2), pp 210–251, (2002) 15˙chapter-15 November 23, 2010 13:50 World Scientific Review Volume - 9in x 6in A Unifying Discontinuous Formulation for Hybrid Meshes 15˙chapter-15 453 41 Z J Wang, High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Progress in Aerospace Sciences 43, pp 1–47, (2007) 42 Z J Wang and H Gao, A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference mthods for conservation laws on mixed grids, J Comput Phys 228, pp 8161–8186, (2009) 43 Z J Wang and Y Liu, Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation, J Comput Phys 179, pp 665–697, (2002) 44 T Warburton, An explicit construction of interpolation nodes on the simplex, J Eng Math 56 (2), pp 247–262, (2006) 45 O C Zienkiewicz and R L Taylor, The Finite Element Method The Basics, vol Butterworth-Heinemann, Oxford, England, (2000) This page intentionally left blank Index adaptive mesh refinement, 67–92 adjoint-based, 70-72 anisotropic, 77 goal-oriented, see adjoint-based, 72 hp-refinement, 74-77 output-based, see adjoint-based, 72 Additive Schwarz, 39 adjoint problem, 70 72, 77 advection, 332, 333, 337, 341, 344, 350, 357 advection-diffusion equation, 95 aeroacoustics, 137 analytical differentiation, 147 approximate Riemann solver, 127 Arbitrary-Lagrangian Eulerian (ALE), 122 artificial diffusion, 128 coarse scale correction, 135 common derivative, 397, 412 common flux, 392, 397, 399, 408 Common values, 411 Compact discontinuous Galerkin (CDG), 126, 438 compressible Navier-Stokes equations, 120 conservation laws, 391, 393, 420, 422 continuous extension Runge-Kutta (CERK), 104 continuous flux function, 396, 400 corrected derivative (estimate), 411 corrected second derivative, 412, 414 correction function, 393, 398, 403, 404, 407, 410, 413, 416, 417, 419 correction function, 399, 401, 407, 418 Correction procedure via reconstruction (CPR), 391, 424 correction terms, 391, 392, 393 CPR Chain rule (CR) approach, 428 correction field, 426 flux points (FPs), 427 Lagrange polynomial (LP) approach, 428 lifting constants, 427 DG coefficients, 430 SD coefficients, 431 SV coefficients, 431 lifting operator, 426 penalty term, 426 Riemann flux, 425 Backward Differentiation Formulas (BDF), 131 basis functions, 392, 395, 396, 405, 408 Bassi-Rebay-2, 196 Bassi-Rebay (BR2), 436 Baumann, 188 BDD, 52 BDDC, 53 boundary terms, 187 BR2, 414, 419 Cauchy problem, 103 CDG, 412, 414, 419, 420, 422 CFL number, 364, 368 Classical Substructuring Methods, 48 455 456 solution points (SPs), 427 standard element, 431 weighted residual formulation, 425 CPR algorithm, 400 deformation gradient, 122 Degrees-of-freedom (DOF), 424 Delaunay refinement, 143 derivative approximation, 299, 300, 301, 304, 321, 327 derivative matrix, 396 DG discretization, BR2 scheme, DG equations once partially integrated, 187 twice partially integrated, 198 DG method, 67–92, 153–175 Diagonally Implicit Runge-Kutta (DIRK) methods, 131, 139, 147 differential form, 391, 392, 423 differential quadrature (DQ), 299, 300, 304 diffusion equation, 393, 411 discontinuity sensor, 130 discontinuous flux function, 395, 397, 399, 400 discontinuous Galerkin, 365, 366, 421, 424 discontinuous Galerkin formulation strong, 100 ultra weak, 99 weak, 100 Discontinuous Galerkin schemes for diffusion, 185, 186 (σ,µ) family of, 186 history, 186 dissipation, 333, 338, 343 DistMesh mesh generator, 143 domain decomposition, 137 eddy viscosity, 121 Efficient scheme, 195 Index eigenvalue, 333, 334, 336, 337, 341, 343, 344, 352, 414–417, 420, 419 eigenvalues and stability, 191 high-frequency accuracy, 191 low-frequency accuracy, 191 ENO schemes, 241 error evolution, 193 high-frequency, 194 low-frequency, 193 of initial projection, 194 error estimation adjoint-based, 70 72 multiple target quantities, 71 single target quantity, 70 eigenvectors, 193 amplitude, 193 and initial projection error, 194 Euler equations, 365 exact dispersion relation, 332 explicit time integration, 95 FAS, see multigrid, 377 FETI, 52 FETI-DP, 53 finite element spaces, 126 flapping wings, 143 flow past a flat plate, 142 flux function, 126 Flux reconstruction (FR), 423 FMG, see multigrid, 382 Fourier analysis, 188, 404, 406, 414, 416, 420, 421 fourth-order schemes, 193 function approximation, 299, 300, 301, 307, 311, 315, 327 Gauss points, 402, 406, 407 Gauss-Seidel, 373 LU-SGS, 374 symmetric, 373 Index Geometric Conservation Law (GCL), 125 global element, 394 GMRES, 132, 249 371 preconditioning, 250, 260 flexible, 376 Godunov method, 423 gradient consistency, 193 and initial projection error, 193 high order, 301, 305, 327 hybrid multilevel schemes, 381 HWENO reconstruction, 153–175 I-continuous, 436 ILU, 376 Implicit Large Eddy Simulation (ILES), 121, 139 implicit time integration, 13 CFL evolution via pseudotransient continuation strategy, 15 linearly implicit Rosenbrock-type Runge-Kutta schemes, 13 time step restriction, 15 incomplete LU (ILU) factorization, 133 incompressible, 299, 317, 323, 328, 316 Inconsistent scheme, 188 interface correction, 398 interior integral, 187 improvement by recovery, 196 inaccuracy of, 196 Interior penalty, 437 Jacobi method, 133 Jacobian, 370 CPU time, 259, 261 explicit, 249 250 high-order, 249 memory usage, 261 preconditioning, 260 Jacobian matrix, 131 457 jump operator, 127 Kelvin-Helmholtz instability, 137 Koornwinder basis, 130 Krylov methods, 371 Krylov subspace methods, 132 Lagrange interpolation, 427, 428, 440 Laminar boundary layer, 445 laminar separation bubble, 139 Laplacian diffusion, 129 Large Eddy Simulation (LES), 121 LDG, 197,412, 414, 419, 420 Lebesgue, 335, 341 Legendre polynomial, 402, 403, 406, Legendre polynomials, 130 Lifting collocation penalty (LCP), 423 limiter, 153–175 limiting, 239 accuracy, 245, 252, 254 at boundaries, 247 Barth Jespersen, 242 convergence, 257 high-order, 243, 244 monotonicity, 256 Venkatakrishnan, 244 linear vector space, 307, 308, 309 load balancing, 136 Lobatto points, 394, 395, 403, 405, 407 Lobatto polynomial, 403 Local Discontinuous Galerkin (LDG) method, 127 local element, 394 local time stepping, 107 lumping for Lobatto points, 405 mapping velocity, 122 mass matrix, 131 matrix-free methods, 374 mesh-free method, 299, 300 method of Lines, 366 458 Minimum Discarded Fill (MDF) method, 134 minimum dissipation scheme, 127 modified dispersion relation, 333, 334, 336, 341, 346 MQ-DQ, 299, 312, 313, 315, 316, 323, 324, 325, 326, 327, 328 multigrid, 377 FAS, 377 FMG, 382 geometric, 377 multigrid method, 135 multi-p methods, 379 linear, 380 Multiplicative Schwarz, 39 multiquadrics (MQ), 299, 302, 303 NACA0012 airfoil, 443 Navier-Stokes equations, 120 Neumann-Neumann Methods, 51 numerical results, 199 accuracy of cell average, 199 accuracy of gradient, 200 Newton iteration, 370 Newton-Krylov method, 131, 147 Newton's method, 131 nodal basis, 131 nonlinear interactions, 137 Nonoverlapping Methods, 45 non-overlapping Schwartz, 137 Numerical Accuracy, 313 numerical discretization, 301, 310 numerical flux function, 127 optimal, 34 optimality, 34 order of accuracy, 404, 415, 418 orthonormal basis functions, 10 modified Gram-Schmidt (MGS) orthogonalization, 11 Overlapping Methods, 37 Index panel method, 148 partial differential equations (PDEs)., 299 perfect gas, 365 Piola relationships, 125 PNPM, 203–233 Poor Man's recovery scheme, 197 potential flow, 148 preconditioning, 132, 250 predictor-corrector formulation, 106 pressure coefficient, 139 principal eigenvalue, 415, 416 prolongation operator, 136 propagation direction, 333, 334, 347 q-criterion, 139 quadrilateral, 333, 354–357, 361 quasi-optimal, 49 Radau points, 403 Radau polynomial, 393, 399, 402, 403, 404, 407, 409 radial basis functions (RBFs), 299, 301 RBF-DQ, 299, 301, 307, 309, 310, 311, 313, 316, 317, 318, 319, 320, 321, 323, 327 RDG-1x, 195 RDG-2x, 197 reattachment, 139 reconstruction, 237, 391, 392, 393, 412, 420, 422 least-squares, 237 239 conditioning, 239 recovery, 193 improves interior integral, 196 Poor Man's, 197 principle, 196 RDG-1x, 195 RDG-2x, 197 Residual distribution schemes: accuracy, 275 approximation space, 272 Index boundary conditions, 280 connections with finite volume methods, 270 elimination of spurious modes, 277 spurious modes, 277 sub-residuals, 274 total residual, 273 viscous flows, 293 residual vector, 131 restriction operator, 136 Reynolds Averaged Navier-Stokes (RANS), 121 Riemann 334, 348–352, 354–356, 360, 361 Ringleb's flow, 251 R-K, 344, 345, 358 Robin-Robin, 55 Roe's method, 127 Runge-Kutta, 367 scalability, 34 scalable, 34 Schur Complement Methods, 45 Schwarz Methods, 37 separation, 139, 147 separation bubble, 139 Shape function, 432, 439 shape parameter, 302, 303, 311, 312, 314, 324, 325, 326, 328 shock capturing, 16 sixth-order scheme, 200 skin friction coefficient, 139 smoother, 135 solution points, 394, 395, 400, 405, 406, 407, 411, 412, 414, 415, 416 solution polynomial, 395, 413 solution polynomial, 395 Spalart-Allmaras model, 121, 141 spectra, 417–419, 422 spectral difference, 365, 366, 391, 392, 422 spectral volume, 391, 392, 422 459 spectrum, 415, 417 stable, 332, 333, 337, 338, 341, 343, 345, 352 354, 357, 360, 361 stability, 188, 391, 393, 404, 406, 414, 416, 417, 421, 422 maximum, 195 Stabilized Symmetric scheme, 188 strong form, 408 sub-cell resolution, 129 subgrid scales, 121 Substructuring Methods, 45 supporting points, 314, 315, 316, 321, 324, 326, 327, 328 Symmetric scheme, 188 test function, 408 time step restriction, 96 Tollmien-Schlichting waves, 141 transition, 139 triangular, 333, 335, 337, 347, 348, 352, 355, 356, 361 turbulence model, 121 turbulent dynamic viscosity, 121 turbulent flow, 88, 90, 141 turbulent flows, κ-ω model, governing equations, realizability constraint on ω, wall boundary condition for ω, turoubled-cell indicator, 153–175 TVD, 241, 367 twist scaling factor, 146 unstable, 336, 337, 341, 343, 344, 350 update operator, 188 eigenvalues, 190 eigenvectors, 193 Fourier transform, 190 upwind, 349 352, 354 356, 360, 361 upwind flux, 394, 397, 408 variational formulation, 99 460 Von Neumann analysis, 414 Voronoi, 388 Vortex propagation problem, 441 W cycle, 378 wave number, 333, 334, 351, 414, 415, 420 wave orienteation, 333 weak form’, 408 weighting coefficients, 300, 305, 307, 309, 310, 312, 317, 318, 321, 327 Index WENO reconstruction, 153–175 WENO schemes, 241 (σ,µ) plane, 190 three lines in, 190 (σ,µ)-family, 186 (σ,µ) plane, 190 update equations, 187 .. .Adaptive High-Order Methods in Computational Fluid Dynamics 7792tp.indd 2/9/11 3:15 PM Advances in Computational Fluid Dynamics Editors -in- Chief: Chi-Wang Shu (Brown... Cataloguing -in- Publication Data A catalogue record for this book is available from the British Library ADAPTIVE HIGH-ORDER METHODS IN COMPUTATIONAL FLUID DYNAMICS Advances in Computational Fluid Dynamics. .. China) & Hang Ding (University of California, Santa Barbara, USA) Steven - Adaptive High-Order Methods. pmd 2/1/2011, 11:47 AM Vol Advances in Computational Fluid Dynamics Adaptive High-Order Methods