Massachusetts Institute of Technology

The book based on these lectures is A Student's Guide to
Numerical
Methodspublished by Cambridge University Press, 2015. And available from Amazon.com. |

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S.D.G.

1.1 Exact fitting

1.1.1 Introduction

1.1.2 Representing an exact fitting function linearly

1.1.3 Solving for the coefficients

1.2 Approximate Fitting

1.2.1 Linear Least Squares

1.2.2 SVD and the Moore-Penrose Pseudo-inverse

1.2.3 Smoothing and Regularization

1.3 Tomographic Image Reconstruction

1.4 Efficiency and Nonlinearity

2 Ordinary Differential Equations

2.1 Reduction to first-order

2.2 Numerical Integration of Initial Value Problem

2.2.1 Explicit Integration

2.2.2 Accuracy and Runge-Kutta Schemes

2.2.3 Stability

2.3 Multidimensional Stiff Equations: Implicit Schemes

2.4 Leap-Frog Schemes

3 Two-point Boundary Conditions

3.1 Examples of Two-Point Problems

3.2 Shooting

3.2.1 Solving two-point problems by initial-value iteration

3.2.2 Bisection

3.3 Direct Solution

3.3.1 Second-order finite differences

3.3.2 Boundary Conditions

3.4 Conservative Differences, Finite Volumes

4 Partial Differential Equations

4.1 Examples of Partial Differential Equations

4.1.1 Fluid Flow

4.1.2 Diffusion

4.1.3 Waves

4.1.4 Electromagnetism

4.2 Classification of Partial Differential Equations

4.3 Finite Difference Partial Derivatives

5 Diffusion. Parabolic Partial Differential Equations

5.1 Diffusion

5.2 Time Advance Choices and Stability

5.2.1 Forward Time, Centered Space

5.2.2 Backward Time, Centered Space. Implicit Scheme.

5.2.3 Partially implicit, Crank-Nicholson schemes.

5.3 Implicit Advancing Matrix Method

5.4 Multiple Space Dimensions

5.5 Estimating Computational Cost

6 Elliptic Problems and Iterative Matrix Solution

6.1 Elliptic Equations and Matrix Inversion

6.2 Convergence Rate

6.3 Successive Over-Relaxation

6.4 Iteration and Nonlinear Equations

6.4.1 Linearization

6.4.2 Combining linear and nonlinear iteration

7 Fluid Dynamics and Hyperbolic Equations

7.1 The fluid momentum equation

7.2 Hyperbolic Equations

7.3 Finite Differences and Stability

7.3.1 FTCS is unstable

7.3.2 Lax-Friedrichs and the CFL condition

7.3.3 Lax-Wendroff achieves second order accuracy

7.4 Worked Example: Three-dimensional fluids

8 Boltzmann's Equation and its solution

8.1 The Distribution Function

8.2 Conservation of Particles in Phase-Space

8.3 Solving the Hyperbolic Boltzmann Equation

8.3.1 Integration along orbits

8.3.2 Orbits are Characteristics

8.4 Collision Term

8.4.1 Self-scattering

8.4.2 No self-scattering

9 Energy-Resolved Diffusive Transport

9.1 Collisions of Neutrons

9.2 Reduction to multigroup diffusion equations

9.3 Numerical Representation of Multigroup Equations

9.3.1 Groups

9.3.2 Steady State Eigenvalue

10 Atomistic and Particle-in-Cell Simulation

10.1 Atomistic Simulation

10.1.1 Atomic/Molecular Forces and Potentials

10.1.2 Computational requirements

10.2 Particle in Cell Codes

10.2.1 Boltzmann Equation Pseudoparticle Representation

10.2.2 Direct Simulation Monte Carlo treatment of gas

10.2.3 Particle Boundary Conditions

11 Monte Carlo Techniques

11.1 Probability and Statistics

11.1.1 Probability and Probability Distribution

11.1.2 Mean, Variance, Standard Deviation, and Standard Error

11.2 Computational random selection

11.3 Flux integration and injection choice.

12 Monte Carlo Radiation Transport

12.1 Transport and Collisions

12.1.1 Random walk step length

12.1.2 Collision type and parameters

12.1.3 Iteration and New Particles

12.2 Tracking, Tallying and Statistical Uncertainty

12.3 Worked Example: Density and Flux Tallies

13 Next Steps

13.1 Finite Element Methods

13.2 Discrete Fourier Analysis and Spectral Methods

13.3 Sparse Matrix Iterative Krylov Solution

13.4 Fluid Evolution Schemes

13.4.1 Incompressible fluids and pressure correction

13.4.2 Nonlinearities, shocks, upwind and limiter differencing

13.4.3 Turbulence

A Summary of Matrix Algebra

A.1 Vector and Matrix Multiplication

A.2 Determinants

A.3 Inverses

A.4 Eigenanalysis

Index

- Become familiar with computational engineering and its mathematical foundations, at an elementary level.
- Deepen their understanding of the basic equations governing the physical phenomena.
- Understand the methods by which problems can be solved using computation.
- Develop experience, confidence, and good critical judgement in the application of numerical methods to the solution of physical problems.
- Strengthen their ability to use computation in theoretical analysis and experimental data interpretation.

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