Computational Partial Differential Equations

Table of contents

Getting Started
The First Diffpack Encounter
What is Diffpack?
A Trivial C++ Program
A Trivial Diffpack Program
Steady 1D Heat Conduction
The Physical and Mathematical Model
A Finite Difference Method
Implementation in Diffpack
Dissection of the Program
Tridiagonal Matrices
Variable Coefficients
A Nonlinear Heat Conduction Problem
Simulation of Waves
Modeling Vibrations of a String
A Finite Difference Method
Implementation
Visualizing the Results
A 2D Wave Equation with Variable Wave
Velocity
A Modelfor Water Waves
Projects
A Uni-Directional Wave Equation
Centered Differences for a Boundary-Layer Problem
Upwind Differences for a Boundary-LayerProblem
About Programming with Objects
Motivation for the Object Concept
Example: Implementation of a Vector Class
in C++
Arrays in Diffpack
Example: Design of an ODE Solver Environment
Abstractions for Grids and Fields
Coding the PDE Simulator as a Class
Steady 1D Heat Conduction Revisited
Empirical Investigation of the Numerical
Method
Simulation of Waves Revisited
Projects
Transient Flow Between Moving Plates
Transient Channel Flow
Coupled Heat and Fluid Flow
Difference Schemes for Transport Equations
Introduction to Finite Element Discretization
Weighted Residual Methods
Basic Principles
Example: A 1D Poisson Equation
Treatment of Boundary Conditions
Time Dependent Problems
A Wave Equation
A Heat Equation
Finite Elements in One Space Dimension
Piecewise Polynomials
Handling of Essential Boundary Conditions
Direct Computation of the Linear System
Element-By-Element Formulation
Extending the Concepts to Quadratic Elements
Summary of the Elementwise Algorithm
Example: A 1D Wave Equation
The Finite Element Equations
Interpretation of the Discrete Equations
Accuracy and Stability
Naive Implementation
Projects
Heat Conduction with Cooling Law
Retardation of a Well-Bore
Higher-Dimensional Finite Elements
The Bilinear Element and Generalizations
The Linear Triangle
Example: A 2D Wave Equation
Other Two-Dimensional Element Types
Three-Dimensional Elements
Calculation of derivatives
Global Least-Squares Smoothing
Flux Computations in Heterogeneous Media
Convection-Diffusion Equations
A One-Dimensional Model Problem
Multi-Dimensional Equations
Time-Dependent Problems
Analysis of the Finite Element Method
Weak Formulations
Variational Problems
Results for Continuous Problems
Results for Discrete Problems
A Priori Error Estimates
Numerical Experiments
Adaptive Finite Element Methods
Programming of Finite Element Solvers
A Simple Program for the Poisson Equation
Discretization
Basic Parts of a Simulator Class
Increasing the Flexibility
A Generalized Model Problem
Using the Menu System
Creating the Grid Object
Some Visualization Tools
Storing Fields for Later Visualization
Filtering Simres Data
Visualizing Diffpack Data in Plotmtv
Visualizing Diffpack Data in Matlab
Visualizing Diffpack Data in Vtk
Visualizing Diffpack Data in IRIS Explorer
Plotting Fields Along Lines Through the
Domain
Some Useful Diffpack Features
The Menu System
Multiple Loops
Computing Numerical Errors
Functors
Computing Derivatives of Finite Element Fields
Specializing Code in Subclass Solvers
Introducing More Flexibility
Setting Boundary Condition Information in the Grid
Line and Surface Integrals
Simple Mesh Generation Tools
Debugging
Automatic Report Generation
Specializing Code in Subclass Solvers
Overriding Menu Answers in the Program
Estimating Convergence Rates
Axisymmetric Formulations and Cartesian
2D Code
Summary
Adaptive Grids
How to Extend an Existing Simulator
Organization of Refinement Criteria
Example: Corner-Flow Singularity
Transient Problems
Projects
Flow in an Open Inclined Channel
Stress Concentration due to Geometric
Imperfections
Lifting Airfoil
A Convection-Diffusion Solver
A Heat Equation Solver
Discretization
Implementation
A More Flexible Heat Equation Solver
About the Model Problem and the Simulator
Visualization and Animation of
Time-Dependent Data
Variable Time Step Size
Applying a Transient Solver to a Stationary PDE
Handling Simulation and Visualization from a Script
Some Computer Exercises Involving Heat Transfer
Efficient Solution of the Wave Equation
Discretization
Implementation
Extensions of the Model Problem
Flexible Representation of Variable
Coefficients
Nonlinear Problems
Discretization and Solution of Nonlinear PDEs
Finite Difference Discretization
Finite Element Discretization
The Group Finite Element Method
Successive Substitutions
Newton-Raphson's Method
A Transient Nonlinear Heat Conduction Problem
Iteration Methods at the PDE Level
Continuation Methods
Software Tools for Nonlinear Finite Element Problems
A Solver for a Nonlinear Heat Equation
Extending the Solver
Projects
Operator Splitting for a
Reaction-Diffusion Model
Compressible Potential Flow
Solid Mechanics Applications
Linear Thermo-Elasticity
Physical and Mathematical Problem
A Finite ElementMethod
Engineering Finite Element Notation
Implementation
Examples
Elasto-Viscoplasticity
Basic Physical Features of Elasto-Viscoplasticity
A Three-Dimensional Elasto-Viscoplastic
Model
Simplification in Case of a Forward
Scheme in Time
Numerical Handling of Yield Criteria
Implementation
Fluid Mechanics Applications
Convection-Diffusion Equations
The Physical and Mathematical Model
A Finite Element Method
Incorporation of Nonlinearities
Software Tools
Melting and Solidification
Shallow Water Equations
The Physical and Mathematical Model
Finite Difference Methods on Staggered Grids
Implementation
Nonlinear and Dispersive Terms
Finite Element Methods
An Implicit Finite Element Navier-Stokes Solver
The Physical and Mathematical Model
A Finite Element Method
Solution of the Nonlinear Systems
Implementation
A Classical Finite Difference Navier-Stokes Solver
Operator Splitting
Finite Differences on 3D Staggered Grids
A Multigrid Solver for the Pressure Equation
Implementation
A Fast Finite Element Navier-Stokes Solver
Operator Splitting and Finite Element
Discretization
An Optimized Implementation
Projects
Analysis of Discrete Shallow Water Waves
Approximating the Navier-Stokes Equations
by a Laplace Equation
Coupled Problems
Fluid-Structure Interaction; Squeeze-Film
Damping
The Physical and Mathematical Model
Numerical Methods
Implementation
Fluid Flow and Heat Conduction in Pipes
The Physical and Mathematical Model
Numerical Methods
Implementation
Projects
Transient Thermo-Elasticity
Convective-Diffusive Transport in Viscous Flow
Chemically Reacting Fluid
Mathematical Topics
Scaling and Dimensionless Variables
Indicial Notation
Compact Notation for Difference Equations
Stability and Accuracy of Difference Approximations