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E-raamat: Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow

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Discontinuous Galerkin Method: Analysis and Applications to Compressible FlowSuurem pilt
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The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.

1 Introduction
1(26)
1.1 DGM Versus Finite Volume and Finite Element Methods
2(3)
1.2 A Short Historical Overview of the DGM
5(4)
1.2.1 DGM for Hyperbolic and Singularly Perturbed Problems
5(1)
1.2.2 DGM for Elliptic and Parabolic Problems
6(2)
1.2.3 DGM for the Numerical Solution of Compressible Flow
8(1)
1.2.4 Monographs Dealing with the DGM
9(1)
1.3 Some Mathematical Concepts
9(18)
1.3.1 Spaces of Continuous Functions
10(1)
1.3.2 Lebesgue Spaces
11(1)
1.3.3 Sobolev Spaces
12(1)
1.3.4 Theorems on Traces and Embeddings
13(3)
1.3.5 Bochner Spaces
16(3)
1.3.6 Useful Theorems and Inequalities
19(8)
Part I Analysis of the Discontinuous Galerkin Method
2 DGM for Elliptic Problems
27(58)
2.1 Model Problem
27(1)
2.2 Abstract Numerical Method and Its Theoretical Analysis
28(3)
2.3 Spaces of Discontinuous Functions
31(6)
2.3.1 Partition of the Domain
31(2)
2.3.2 Assumptions on Meshes
33(2)
2.3.3 Broken Sobolev Spaces
35(2)
2.4 DGM Based on a Primal Formulation
37(6)
2.5 Basic Tools of the Theoretical Analysis of DGM
43(8)
2.5.1 Multiplicative Trace Inequality
46(2)
2.5.2 Inverse Inequality
48(1)
2.5.3 Approximation Properties
49(2)
2.6 Existence and Uniqueness of the Approximate Solution
51(10)
2.6.1 The Choice of Penalty Weight σ
52(1)
2.6.2 Continuity of Diffusion Bilinear Forms
53(6)
2.6.3 Coercivity of Diffusion Bilinear Forms
59(2)
2.7 Error Estimates
61(7)
2.7.1 Estimates in the DG-Norm
61(3)
2.7.2 Optimal L2(Ω)-Error Estimate
64(4)
2.8 Baumann--Oden Method
68(4)
2.9 Numerical Examples
72(13)
2.9.1 Regular Solution
72(5)
2.9.2 Singular Case
77(1)
2.9.3 A Note on the L2(Ω)-Optimality of NIPG and IIPG
78(7)
3 Methods Based on a Mixed Formulation
85(32)
3.1 A General Mixed DG Method
85(5)
3.1.1 Equivalent Formulations
87(1)
3.1.2 Lifting Operators
88(2)
3.2 Bassi--Rebay Methods
90(14)
3.2.1 Mixed Formulation
90(1)
3.2.2 Variational Formulation
91(4)
3.2.3 Theoretical Analysis
95(9)
3.3 Local Discontinuous Galerkin Method
104(13)
3.3.1 Mixed Formulation
105(3)
3.3.2 Variational Formulation
108(2)
3.3.3 Theoretical Analysis
110(7)
4 DGM for Convection-Diffusion Problems
117(54)
4.1 Scalar Nonlinear Nonstationary Convection-Diffusion Equation
117(3)
4.2 Discretization
120(4)
4.3 Abstract Error Estimate
124(14)
4.3.1 Consistency of the Convection Form in the Case of the Dirichlet Boundary Condition
125(3)
4.3.2 Consistency of the Convective Form in the Case of Mixed Boundary Conditions
128(6)
4.3.3 Error Estimates for the Method of Lines
134(4)
4.4 Error Estimates in Terms of h
138(4)
4.5 Optimal L∞(0, T; L2(Ω))-Error Estimate
142(9)
4.6 Uniform Error Estimates with Respect to the Diffusion Coefficient
151(15)
4.6.1 Continuous Problem
151(2)
4.6.2 Discretization of the Problem
153(4)
4.6.3 Error Estimates
157(9)
4.7 Numerical Examples
166(5)
5 Space-Time Discretization by Multistep Methods
171(52)
5.1 Semi-implicit Backward Euler Time Discretization
171(12)
5.1.1 Discretization of the Problem
172(1)
5.1.2 Error Estimates
173(10)
5.2 Backward Difference Formula for the Time Discretization
183(40)
5.2.1 Discretization of the Problem
184(4)
5.2.2 Theoretical Analysis
188(10)
5.2.3 Error Estimates
198(23)
5.2.4 Numerical Examples
221(2)
6 Space-Time Discontinuous Galerkin Method
223(114)
6.1 Space-Time DGM for a Heat Equation
223(44)
6.1.1 Discretization of the Problem
224(2)
6.1.2 Space-Time DG Discretization
226(3)
6.1.3 Auxiliary Results
229(2)
6.1.4 Space-Time Projection Operator
231(10)
6.1.5 Abstract Error Estimate
241(4)
6.1.6 Estimation of Projection Error in Terms of h and τ
245(8)
6.1.7 Error Estimate in the DG-norm
253(2)
6.1.8 Discrete Characteristic Function
255(6)
6.1.9 Error Estimate in the L∞ (0, T; L2(Ω))-norm
261(3)
6.1.10 The Case of Identical Meshes on All Time Levels
264(1)
6.1.11 Alternative Proof of Lemma 6.12
264(3)
6.2 Space-Time DGM for Nonlinear Convection-Diffusion Problems
267(26)
6.2.1 Discretization of the Problem
268(2)
6.2.2 Auxiliary Results
270(9)
6.2.3 Abstract Error Estimate
279(10)
6.2.4 Main Result
289(3)
6.2.5 Numerical Examples
292(1)
6.3 Extrapolated Space-Time Discontinuous Galerkin Method for Nonlinear Convection-Diffusion Problems
293(24)
6.3.1 Discretization of the Problem
294(3)
6.3.2 Auxiliary Results
297(7)
6.3.3 Error Estimates
304(7)
6.3.4 Numerical Examples
311(6)
6.4 Uniform Error Estimates with Respect to the Diffusion Coefficient for the ST-DGM
317(20)
6.4.1 Formulation of the Problem and Some Assumptions
319(1)
6.4.2 Discretization of the Problem
319(3)
6.4.3 Properties of the Discrete Problem
322(1)
6.4.4 Abstract Error Estimate
323(9)
6.4.5 Numerical Examples
332(5)
7 Generalization of the DGM
337(64)
7.1 hp-Discontinuous Galerkin Method
337(24)
7.1.1 Formulation of a Model Problem
338(1)
7.1.2 Discretization
338(4)
7.1.3 Theoretical Analysis
342(10)
7.1.4 Computational Performance of the hp-DGM
352(9)
7.2 DGM on General Elements
361(13)
7.2.1 Assumptions on the Domain Partition
362(1)
7.2.2 Function Spaces
363(1)
7.2.3 Approximate Solution
364(1)
7.2.4 Auxiliary Results
365(5)
7.2.5 Error Analysis
370(1)
7.2.6 Numerical Examples
370(4)
7.3 The Effect of Numerical Integration
374(27)
7.3.1 Continuous Problem
374(1)
7.3.2 Space Semidiscretization
375(1)
7.3.3 Numerical Integration
376(1)
7.3.4 Some Important Results
377(2)
7.3.5 Truncation Error of Quadrature Formulae
379(4)
7.3.6 Properties of the Convection Forms
383(4)
7.3.7 The Effect of Numerical Integration in the Convection Form
387(5)
7.3.8 Error Estimates for the Method of Lines with Numerical Integration
392(9)
Part II Applications of the Discontinuous Galerkin Method
8 Inviscid Compressible Flow
401(76)
8.1 Formulation of the Inviscid Flow Problem
402(7)
8.1.1 Governing Equations
402(6)
8.1.2 Initial and Boundary Conditions
408(1)
8.2 DG Space Semidiscretization
409(4)
8.2.1 Notation
409(2)
8.2.2 Discontinuous Galerkin Space Semidiscretization
411(2)
8.3 Numerical Treatment of Boundary Conditions
413(10)
8.3.1 Boundary Conditions on Impermeable Walls
413(3)
8.3.2 Boundary Conditions on the Inlet and Outlet
416(7)
8.4 Time Discretization
423(24)
8.4.1 Backward Euler Method
424(1)
8.4.2 Newton Method Based on the Jacobi Matrix
425(1)
8.4.3 Newton-Like Method Based on the Flux Matrix
426(6)
8.4.4 Realization of the Iterative Algorithm
432(2)
8.4.5 Higher-Order Time Discretization
434(5)
8.4.6 Choice of the Time Step
439(2)
8.4.7 Structure of the Flux Matrix
441(2)
8.4.8 Construction of the Basis in the Space Shp
443(2)
8.4.9 Steady-State Solution
445(2)
8.5 Shock Capturing
447(9)
8.5.1 Jump Indicators
448(1)
8.5.2 Artificial Viscosity Shock Capturing
449(2)
8.5.3 Numerical Examples
451(5)
8.6 Approximation of a Nonpolygonal Boundary
456(11)
8.6.1 Curved Elements
456(2)
8.6.2 DGM Over Curved Elements
458(5)
8.6.3 Numerical Examples
463(4)
8.7 Numerical Verification of the BDF-DGM
467(10)
8.7.1 Inviscid Low Mach Number Flow
467(2)
8.7.2 Low Mach Number Flow at Incompressible Limit
469(3)
8.7.3 Isentropic Vortex Propagation
472(2)
8.7.4 Supersonic Flow
474(3)
9 Viscous Compressible Flow
477(44)
9.1 Formulation of the Viscous Compressible Flow Problem
477(8)
9.1.1 Governing Equations
477(6)
9.1.2 Initial and Boundary Conditions
483(2)
9.2 DG Space Semidiscretization
485(7)
9.2.1 Notation
485(1)
9.2.2 DG Space Semidiscretization of Viscous Terms
486(5)
9.2.3 Semidiscrete Problem
491(1)
9.3 Time Discretization
492(5)
9.3.1 Time Discretization Schemes
492(1)
9.3.2 Solution Strategy
493(4)
9.4 Numerical Examples
497(24)
9.4.1 Blasius Problem
498(6)
9.4.2 Stationary Flow Around the NACA 0012 Profile
504(6)
9.4.3 Unsteady Flow
510(2)
9.4.4 Steady Versus Unsteady Flow
512(2)
9.4.5 Viscous Shock-Vortex Interaction
514(7)
10 Fluid-Structure Interaction
521(32)
10.1 Formulation of Flow in a Time-Dependent Domain
521(10)
10.1.1 Space Discretization of the Flow Problem
523(5)
10.1.2 Time Discretization by the BDF Method
528(2)
10.1.3 Space-Time DG Discretization
530(1)
10.2 Fluid-Structure Interaction
531(22)
10.2.1 Flow-Induced Airfoil Vibrations
531(6)
10.2.2 Interaction of Compressible Flow and an Elastic Body
537(16)
References 553(14)
Index 567
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