Title | Advanced Topics in Computational Partial Differential Equations [electronic resource] : Numerical Methods and Diffpack Programming / edited by Hans Petter Langtangen, Aslak Tveito |
---|---|
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003 |
Connect to | http://dx.doi.org/10.1007/978-3-642-18237-2 |
Descript | XIX, 663 p. 13 illus. online resource |
1 Parallel Computing -- 1.1 Introduction to Parallel Computing -- 1.2 A Different Performance Model -- 1.3 The First MPI Encounter -- 1.4 Basic Parallel Programming with Diffpack -- 1.5 Parallelizing Explicit FD Schemes -- 1.6 Parallelizing FE Computations on Unstructured Grids -- References -- Overlapping Domain Decomposition Methods -- 2.1 Introduction -- 2.2 The Mathematical Formulations -- 2.3 A 1D Example -- 2.4 Some Important Issues -- 2.5 Components of Overlapping DD Methods -- 2.6 A Generic Implementation Framework -- 2.7 Parallel Overlapping DD Methods -- 2.8 Two Application Examples -- References -- 3 Software Tools for Multigrid Methods -- 3.1 Introduction -- 3.2 Sketch of How Multilevel Methods are Implemented in Diffpack -- 3.3 Implementing Multigrid Methods -- 3.4 Setting up an Input File -- 3.5 Playing Around with Multigrid -- 3.6 Equipping the Poisson2 Solver with Multigrid -- 3.7 Systems of Equations, Linear Elasticity -- 3.8 Nonlinear Problems -- References -- 4 Mixed Finite Elements -- 4.1 Introduction -- 4.2 Model Problems -- 4.3 Mixed Formulation -- 4.4 Some Basic Concepts of a Finite Element -- 4.5 Some Code Examples -- 4.6 Programming with Mixed Finite Elements in a Simulator -- References -- 5 Systems of PDEs and Block Preconditioning -- 5.1 Introduction -- 5.2 Block Preconditioners in General -- 5.3 The Bidomain Equations -- 5.4 Two Saddle Point Problems -- References -- 6 Fully Implicit Methods for Systems of PDEs -- 6.1 Introduction -- 6.2 Implementation of Solvers for PDE Systems in Diffpack -- 6.3 Problem with the Gauss-Seidel Method, by Example -- 6.4 Fully Implicit Implementation -- 6.5 Applications -- 6.6 Conclusion -- References -- 7 Stochastic Partial Differential Equations -- 7.1 Introduction -- 7.2 Some Simple Examples -- 7.3 Solution Methods -- 7.4 Quick Overview of Diffpack Tools -- 7.5 Tools for Random Variables -- 7.6 Diffpack Tools for Random Fields -- 7.7 Summary -- 7.A Transformation of Random Variables -- 7.B Implementing a New Distribution -- References -- 8 Using Diffpack from Python Scripts -- 8.1 Introduction -- 8.2 Developing Python Interfaces to C/C++ Functions -- 8.3 Compiling and Linking Wrapper Code with Diffpack -- 8.4 Converting Data between Diffpack and Python -- 8.5 Building an Interface to a More Advanced Simulator -- 8.6 Installing Python, SWIG etc -- 8.7 Concluding Remarks -- References -- 9 Performance Modeling of PDE Solvers -- 9.1 Introduction -- 9.2 Model Problems -- 9.3 Numerical Methods -- 9.4 Total CPU Time Consumption -- 9.5 Solution of Linear Systems -- 9.6 Construction of Linear Systems -- 9.7 Concluding Remarks -- References -- 10 Electrical Activity in the Human Heart -- 10.1 The Basic Physiology -- 10.2 Outline of a Mathematical Model -- 10.3 The Bidomain Model -- 10.4 A Complete Mathematical Model -- 10.5 Physiology of the Heart Muscle Tissue -- 10.6 The Numerical Method -- 10.7 Implementation -- 10.8 Optimization of the Simulator -- 10.9 Simulation Results -- 10.10 Concluding Remarks -- References -- 11 Mathematical Models of Financial Derivatives -- 11.1 Introduction -- 11.2 Basic Assumptions -- 11.3 Forwards and Futures -- 11.4 The Black-Scholes Analysis -- 11.5 European Call and Put Options -- 11.6 American Options -- 11.7 Exotic Options -- 11.8 Hedging -- 11.9 Remarks -- References -- 12 Numerical Methods for Financial Derivatives -- 12.1 Introduction -- 12.2 Model Summary -- 12.3 Monte-Carlo Methods -- 12.4 Lattice Methods -- 12.5 Finite Difference Methods -- 12.6 Finite Element Methods -- References -- 13 Finite Element Modeling of Elastic Structures -- 13.1 Introduction -- 13.2 An Introductory Example; Bar Elements -- 13.3 Another Example; Beam Elements -- 13.4 General Three-Dimensional Elasticity -- 13.5 Degrees of Freedom and Basis Functions -- 13.6 Material Types and Elasticity Matrices -- 13.7 Element Matrices in Local Coordinates -- 13.8 Element Load Vectors in Local Coordinates -- 13.9 Element Matrices and Vectors in Global Coordinates -- 13.10 Element Forces, Stresses, and Strains -- 13.11 Implementation of Structural Elements -- 13.12 Some Example Programs -- 13.13 Test Problems -- 13.14 Summary -- References -- 14 Simulation of Aluminum Extrusion -- 14.1 Introduction -- 14.2 Mathematical Formulation -- 14.3 Finite Element Implementation -- 14.4 Object-Oriented Implementation -- 14.5 Numerical Experiments -- 14.6 Concluding Remarks -- References -- 15 Simulation of Sedimentary Basins -- 15.1 Introduction -- 15.2 The Geomechanical and Mathematical Problem -- 15.3 Numerical Methods -- 15.4 Implementing a Solver for a System of PDEs -- 15.5 Verification -- 15.6 A Magmatic Sill Intrusion Case Study -- 15.7 Concluding Remarks -- References