Numerical Mathematics and Advanced Applications ENUMATH 2015

Name: Numerical Mathematics and Advanced Applications ENUMATH 2015
Brand: Springer
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Bülent Karasözen Murat Manguoglu Münevver Tezer-Sezgin Serdar Göktepe Ömür Ugur(Editor)

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Published on 9. November 2016

XIV, 643 pages

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Content

Intro
Preface
Contents
Part I Space Discretization Methods for PDEs
DRBEM Solution of MHD Flow and Electric Potential in a Rectangular Pipe with a Moving Lid
1 Introduction
2 The Physical Problem and Mathematical Formulation
3 DRBEM Application
4 Numerical Results
5 Conclusion
References
DRBEM Solution of the Double Diffusive Convective Flow
1 Introduction
2 Governing Equations
3 Application of the DRBEM
4 Numerical Results
5 Conclusion
References
Complete Flux Scheme for Conservation Laws Containing a Linear Source
1 Introduction
2 Modification of the Homogeneous Flux
3 Modification of the Inhomogeneous Flux
4 Numerical Example
5 Concluding Remarks
References
Second Order Implicit Schemes for Scalar Conservation Laws
1 Introduction
2 Different Space Discretizations
2.1 Finite Element Method
2.2 Finite Volume Method with Flux-Limiting
2.3 Discontinuous Galerkin Method
3 Discretization in Time
4 Numerical Examples
4.1 Transport Equation
4.2 Buckley-Leverett Equation
5 Conclusion
References
Flux Approximation Scheme for the Incompressible Navier-Stokes Equations Using Local Boundary Value Problems
1 Introduction
2 Finite Volume Method
3 Integral Representation of the Fluxes
3.1 Closure of the Scheme
4 Numerical Results
5 Conclusion
References
On the Full and Global Accuracy of a Compact Third Order WENO Scheme: Part II
1 Introduction
2 Numerical Scheme
2.1 Reconstruction Procedure
2.2 Fully Discrete Scheme
3 New Results
3.1 A Sufficient Condition in the Linear Case
3.2 Failure of q&2: Numerical Evidence
3.3 Failure of q&2: Analytical Evidence
4 Conclusion
References
The Application of the Boundary Element Method to the Theory of MHD Faraday Generators
1 The MHD Faraday Generator
2 The Boundary Value Problem
2.1 The Equations
2.2 Boundary and Matching Conditions
3 Singular Equations
3.1 The Unknown Functions
4 Discretization
4.1 The Linear System
5 The Velocity and the Electric Current
6 The Output Power
References
Part II Finite Element Methods
How to Avoid Mass Matrix for Linear Hyperbolic Problems
1 Introduction
2 Description of the Scheme
2.1 Formulation for Unsteady Problems
3 Numerical Illustrations
3.1 Parameters
3.2 Simulations
4 Conclusions, Perspectives
References
Two-Dimensional H(div)-Conforming Finite Element Spaces with hp-Adaptivity
1 Introduction
2 Approximation Spaces in H(div,O)
3 Application to Mixed Finite Element Formulation
References
Finite Elements for the Navier-Stokes Problem with OutflowCondition
1 Introduction
2 Variational Formulation
3 Finite Element Discretization
4 A Priori Estimate
5 Numerical Results
References
Quasi-Optimality Constants for Parabolic Galerkin Approximation in Space
1 Introduction
2 Petrov-Galerkin Framework and Quasi-Optimality
3 Two Weak Formulations of Linear Parabolic Problems
3.1 Standard Weak Formulation
3.2 Ultra-Weak Formulation
4 Galerkin Approximation in Space and Quasi-Optimality Constants
4.1 Standard Weak Formulation
4.2 Ultra-Weak Formulation
References
Numerical Studies on a Second Order Explicitly Decoupled Variational Multiscale Method
1 Introduction
2 Numerical Scheme
3 Numerical Experiments
3.1 Numerical Experiment 1: Convergence Rate Verification
3.2 Numerical Experiment 2: Marsigli's Flow with Re=2,000
4 Conclusions
References
Numerical Experiments for Multiscale Problems in Linear Elasticity
1 Introduction
2 Finite Element Heterogeneous Multiscale Method for Linear Elasticity
3 A Priori Error Estimates
4 Numerical Experiments
References
The Skeleton Reduction for Finite Element Substructuring Methods
1 Substructuring, Trace Spaces, and Minimization
2 Weakly Conforming Approximation
3 Minimal Residuals
References
Iterative Coupling of Variational Space-Time Methods for Biot's System of Poroelasticity
1 Introduction
2 Iterative Coupling Scheme and Space-Time Approximation of the Subproblems
3 Numerical Experiments
4 Summary
References
Part III Discontinuous Galerkin Methods for PDEs
Discontinuous Galerkin Method for the Solution of Elasto-Dynamic and Fluid-Structure Interaction Problems
1 Description of the Dynamic Elasticity Problem
2 Discrete Problem
3 Numerical Experiments
3.1 A Benchmark Problem
3.2 Example of Fluid-Structure Interaction
4 Conclusion
References
Stable Discontinuous Galerkin FEM Without Penalty Parameters
1 Introduction
2 Notation
3 Stability of Lifted Gradients
4 Counterexample to Stability for Equal-Order Liftings
5 A Modified LDG Method Without Penalty Parameters
6 Conclusions
References
Time-Space Adaptive Method of Time Layers for the Advective Allen-Cahn Equation
1 Introduction
2 Time-Space Discretization
3 Adaptive Method of Time Layers (AMOT)
4 Numerical Experiments
4.1 Sheering Flow
4.2 Expanding Flow
References
Semi-implicit DGM Applied to a Model of Flocking
1 Continuous Problem
2 Discretization
2.1 Discontinuous Galerkin Space Semidiscretization
2.2 Numerical Flux
2.3 Time Discretization
2.3.1 Linearization of the Source Terms lh
3 Numerical Experiment
4 Conclusion
References
Discontinuous and Enriched Galerkin Methods for Phase-Field Fracture Propagation in Elasticity
1 Introduction
2 The Phase-Field Fracture Model
3 A Quasi-monolithic Incremental Formulation
4 Spatial Discretization with DG and EG
5 A Numerical Test: Single Edge Notched Tension
References
Numerical Method Based on DGM for Solving the System of Equations Describing Motion of Viscoelastic Fluid with Memory
1 Introduction
2 Problem Formulation
3 Discretization
3.1 Finite-Dimensional Spaces and Subdivision of Time Intervals
3.2 Discretization of the Navier-Stokes Equations
3.3 Discretization of the Evolution Equation for the Finger Strain Tensor
4 Numerical Experiment
5 Conclusion
References
Stability Analysis of the ALE-STDGM for Linear Convection-Diffusion-Reaction Problems in Time-Dependent Domains
1 Formulation of the Continuous Problem
2 Space-Time Semidiscretization
3 Analysis of the Stability
References
A Posteriori Error Estimates for Nonstationary Problems
1 Introduction
2 Continuous Problem
3 Discretization
4 A Posteriori Error Analysis
4.1 Error Measure
4.2 Reconstruction of the Solution with Respect to Time
4.3 Reconstruction of the Solution with Respect to Space
4.4 Upper Error Bound
4.5 Asymptotic Lower Error Bound
References
Part IV Numerical Linear Algebra and High Performance Computing
Multigrid at Scale?
1 Introduction
2 Multi-level Methods
3 Fault Model
4 Lyapunov Exponents
5 Summary of Results on Convergence
References
A Highly Scalable Implementation of Inexact Nonlinear FETI-DP Without Sparse Direct Solvers
1 Introduction
2 Inexact Nonlinear FETI-DP
3 Model Problems and Numerical Results
References
A Parallel Multigrid Solver for Time-Periodic Incompressible Navier-Stokes Equations in 3D
1 Introduction
2 Governing Equations
3 Discretization
4 Solution Strategy
5 Numerical Experiments
6 Conclusions
References
Discretization and Parallel Iterative Schemes for Advection-Diffusion-Reaction Problems
1 Discretization and Iterative Solution
2 Numerical Results
2.1 A Three-Dimensional Flow Problem
2.2 Parallel Scalability
3 Conclusions
References
A Simple Proposal for Parallel Computation Over Time of an Evolutionary Process with Implicit Time Stepping
1 Introduction
2 Proposal Outline
2.1 Proposed Approach
3 Convection-Diffusion Equation
4 Numerical Results
5 Conclusions
References
The Induced Dimension Reduction Method Applied to Convection-Diffusion-Reaction Problems
1 Introduction
2 Krylov Methods for Solving Systems of Linear Equations
2.1 The Induction Dimension Reduction Method (IDR(s))
3 Numerical Experiments
3.1 IDR(s) and Bi-CG
3.2 IDR(s), GMRES, and Restarted GMRES
3.3 IDR(s) and Bi-CGSTAB
4 Conclusions
References
Block Variants of the COCG and COCR Methods for Solving Complex Symmetric Linear Systems with Multiple Right-Hand Sides
1 Introduction
2 The Block COCG and Block COCR Methods
2.1 Derivation of the Block COCG and Block COCR Methods
2.2 Improving the Numerical Stability of the Block COCG and Block COCR Methods by Residual Orthonormalization
3 Numerical Experiments
4 Conclusions
References
Part V Reduced Order Modeling
Model Reduction for Multiscale Lithium-Ion Battery Simulation
1 Introduction
2 Reduced Basis Methods Applied to Pore-Scale Battery Models
2.1 A Pore-Scale Lithium-Ion Battery Model
2.2 Reduced Basis Method and Empirical Interpolation
2.3 Numerical Experiments
3 Localized Reduced Basis Multiscale Approximation of Heat Conduction
3.1 A Battery: Heat Conduction Model with Resolved Electrode Geometry
3.2 Localization of Reduced Basis Methods: LRBMS
3.3 Numerical Experiments
4 Conclusion
References
Multiscale Model Reduction Methods for Flow in Heterogeneous Porous Media
1 Introduction
2 Multiscale Porous Media and Flow Models
2.1 Two-Scale Porous Media
2.2 Three-Scale Porous Media
3 Numerical Multiscale Methods
3.1 Cell Problems Transformation and Discretization
3.2 General form of a Cell Problem
4 Model-Order Reduction
4.1 Petrov-Galerkin RB Method
4.2 RB Method at the Micro Scale
4.3 RB Method at the Meso Scale
5 Numerical Experiments
References
Output Error Estimates in Reduced Basis Methods for Time-Harmonic Maxwell's Equations
1 Time-Harmonic Maxwell's Equations
2 Reduced Basis Method
3 Numerical Results
References
Reduced Basis Exact Error Estimates with Wavelets
1 Introduction
2 Reduced Basis Method (RBM) Error Estimation
2.1 Residual-Based Error Estimators
2.2 (Theoretical) Computing via Affine Decomposition and Riesz Representation
2.3 The ``Truth''
2.4 Estimating the Exact Error
3 Wavelet-Based Error Estimation
3.1 Wavelet Bases
3.2 Wavelet-Based Residual Expansion
4 Numerical Experiments
References
Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equations
1 Introduction
2 Full Order Model
3 Reduced Order Model
4 Discrete Empirical Interpolation Method (DEIM)
5 Numerical Results
6 Conclusions and Outlook
References
Local Parametrization of Subspaces on Matrix Manifolds via Derivative Information
1 Motivation: Parametric Model Reduction
2 A Taylor-Like Expansion for Stiefel and Grassmann Curves
3 Numerical Illustration
References
Reduced-Order Multiobjective Optimal Control of Semilinear Parabolic Problems
1 Introduction
2 Problem Formulation and Pareto Optimality
3 The Scalar-Valued Optimal Control Problem
4 Numerical Solution Strategy
5 Numerical Example
References
Part VI Problems with Singularities
Coupling Fluid-Structure Interaction with Phase-Field Fracture: Modeling and a Numerical Example
1 Introduction
2 Notation, Spaces, Equations
2.1 Variational-Monolithic ALE Fluid-Structure Interaction
2.2 Variational Phase-Field for Dynamic Pressurized-Fractures
2.3 The Final System
3 Aspects of Discretization and the Solution Algorithm
4 A Prototype Numerical Example
5 Conclusions
References
Weighted FEM for Two-Dimensional Elasticity Problem with Corner Singularity
1 Introduction
2 Notation: Auxiliary Statements
3 Problem Statement: Rv-Generalized Solution
4 Weighted Finite Element Method
5 Results of Numerical Experiments
References
A Local Error Estimate for the Poisson Equation with a Line Source Term
1 Introduction
2 Main Result
3 Proof of Lemma 1
4 Numerical Experiments
References
Multirate Undrained Splitting for Coupled Flow and Geomechanics in Porous Media
1 Introduction
2 Model Equations, Discretization and Splitting Algorithm
2.1 Mixed Variational Formulation
3 Multirate Formulation: Undrained Split Iterative Method
3.1 Undrained Split Multirate Algorithm
4 Analysis of the Multirate Undrained Scheme
4.1 Weak Formulation of Difference of Two Successive Iterates
Step 1: Mechanics Equation
Step 2: Flow Equation
Step 3: Combining Mechanics and Flow
Step 4: Identifying the Parameters
5 Main Result: Contraction
References
Part VII Computational Fluid Dynamics
CFD Simulation of Interaction between a Fluid and a VibratingProfile
1 Introduction
2 Mathematical Description of the Problem
2.1 Governing System of Equations
2.2 Description of the Profile Motion
3 Numerical Solution
3.1 Modified Causon's Scheme for FVM
3.2 Profile Motion Solution
3.3 Mesh Modification for the ALE Computation
3.4 Boundary Conditions
4 Numerical Results
5 Conclusion
References
Chebyshev Spectral Collocation Method for Natural Convection Flow of a Micropolar Nanofluid in the Presence of a Magnetic Field
1 Introduction
2 Governing Equations
3 Numerical Implementation
4 Results and Discussions
5 Conclusion
References
Drag Reduction via Phase Randomization in Turbulent Pipe Flow
1 Introduction
2 Numerical Method
2.1 Basis Functions and Projection
3 Phase Randomization
4 Results and Conclusion
References
CFD Optimization of a Vegetation Barrier
1 Introduction
2 Numerical Model
2.1 Physical Model
2.1.1 Fluid Flow
2.1.2 Turbulence
2.1.3 Particle Transport
2.1.4 Vegetation
2.2 Numerical Methods
2.2.1 CFD Solver
2.2.2 Optimization
3 Application to the Model Problem
3.1 Case Settings
3.2 Results
4 Discussion
References
Modified Newton Solver for Yield Stress Fluids
1 Introduction
2 Non-standard Saddle Point Problem Formulation
3 Robust Nonlinear Solver
4 Summary
References
Numerical Simulation of 3D Flow of Viscous and Viscoelastic Fluids in T-Junction Channel
1 Introduction
2 Mathematical Model
3 Numerical Solution
4 Numerical Results
5 Conclusions
References
Computational Simulations of Fractional Flow Reserve Variability
1 Introduction
2 Methods
2.1 Patient-Specific 1D Coronary Network
2.2 Numerical Implementation
3 Results
4 Discussion
References
On the Mathematical Modeling of Monocytes Transmigration
1 Introduction
2 The Simplified Oldroyd-B Problem
3 Numerical Results
3.1 Setting up the Numerical Model
3.2 Numerical Implementation
3.3 Results
4 Conclusion
References
Part VIII Computational Methods for Multi-physics Phenomena
Parallel Two-Level Overlapping Schwarz Methods in Fluid-Structure Interaction
1 The Two-Level Overlapping Schwarz Preconditioner
2 Monolithic Fluid-Structure Interaction
2.1 Model Description
2.2 Monolithic Coupling in FSI
2.3 Linearization and Parallel Monolithic Preconditioner
3 Numerical Results for Fluid-Structure Interaction
3.1 Time to Solution Using Different Preconditioners for the Structure Block
3.2 Strong Scaling for the Fluid-Structure Interaction Problem
References
Finite Volume Scheme for Modeling of NAPL Vapor Transport in Air
1 Introduction
2 Mathematical Model
3 Numerical Solution
4 Numerical Test
5 Conclusions
References
Numerical Solution of Constrained Curvature Flow for Closed Planar Curves
1 Introduction
2 Parametric Method
3 Tangential Effects
4 Numerical Solution
5 Computational Studies
6 Conclusion
References
Analysis of a T,?-? Formulation of the Eddy Current Problem Based on Edge Finite Elements
1 Introduction
2 T,?-? Formulation of the Eddy Current Problem
3 Mathematical Analysis of the T,?-? Formulation
4 Finite Element Discretization
5 Numerical Results
References
Two Variants of Stabilized Nodal-Based FEM for the Magnetic Induction Problem
1 Introduction
2 Continuous Magnetic Induction Equation
3 Discretization of the Induction Problem
4 Numerical Simulations
5 Summary: Conclusions
References
Modeling of a Three-Dimensional Spherulite Microstructure in Semicrystalline Polymers
1 Introduction
2 Constitutive Models and Finite Element Model
3 Results
References
Numerical Approximation of Interaction of Fluid Flow and Elastic Structure Vibrations
1 Introduction
2 Mathematical Model
2.1 Elastic Structure
2.2 ALE Method
2.3 Fluid Flow
2.4 Coupling Conditions
3 Numerical Model
3.1 Elastic Structure
3.2 Fluid Flow
4 Numerical Results
5 Conclusion
References
Part IX Miscellaneous Topics
Comparison of Nonlocal Operators Utilizing Perturbation Analysis
1 Introduction
2 Operator Definitions
2.1 Boundary Conditions
2.2 An Alternative Governing Operator
3 Perturbation Expansions
3.1 Operator L
3.1.1 OperatorL, x IL, =(-1, -1+d)
3.1.2 OperatorLc, x IL, c=(-1+d,1-d)
3.1.3 OperatorLr, x IL, r=(1-d,1)
3.1.4 Values ofL at Boundary and Transition Points
3.2 Operator M
3.2.1 OperatorM, x IM, =(-1, -1+d)
3.2.2 OperatorMc, x IM, c=(-1+d, 1-d)
3.2.3 OperatorMr, x IM, r=(1-d, 1)
3.2.4 Values ofM at Boundary and Transition Points
3.3 Operator N
3.3.1 Operator N, x IN, =(-1, -d)
3.3.2 Operator Nc, x IN, c=(-d, d)
3.3.3 Operator Nr, x IN, r=(d,1)
3.3.4 Values of N at Boundary and Transition Points
4 Comparison of Operators
4.1 Comparison in the Bulk
4.2 Comparison of Higher Order Approximations in the Bulk
4.3 Comparison at the Boundary and Transition Points
5 Conclusion
References
Pricing of Basket Options Using Dimension Reduction and Adaptive Finite Differences in Space, and Discontinuous Galerkin in Time
1 Introduction
2 Dimension Reduction
3 Adaptive Finite Differences in Space
4 Discontinuous Galerkin in Time
5 Numerical Results
6 Conclusions
References
On the Stability of a Weighted Finite Difference Scheme for Hyperbolic Equation with Integral Boundary Conditions
1 Introduction
2 A Weighted Finite Difference Scheme
2.1 Notation
2.2 Discrete Problem
2.3 The Three-Layer Finite Difference Scheme
2.4 The Two-Layer Finite Difference Scheme
3 Spectrum Analysis
3.1 The Eigenstructure of the Matrix ?
3.2 The Structure of the Spectrum of the Matrix S
References
A Riemannian BFGS Method for Nonconvex Optimization Problems
1 Introduction
2 Notation
3 Riemannian BFGS Method with Cautious Update
4 Convergence Analysis
5 Experiments
5.1 Problem, Retraction, Vector Transport and Step Size
5.2 Tests and Results
6 Conclusion
References
Discrete Lie Derivative
1 Introduction
2 Properties of the Lie Derivative
3 Derivations at a Point
4 The Discrete Lie Derivative of a 1-Cochain
5 Conclusion
References

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Numerical Mathematics and Advanced Applications ENUMATH 2015

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