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Introduction to Scientific Computing
Basic Linear Algebra
1. Overview
2. Development environment
3. Refresh your C++
4. Creating documentation
5. Expression templates
6. Automatic testing
7. Python bindings
8. Interfacing Lapack
Performance
9. Overview
10. Vectorization
10.1. The SIMD - class
10.2. Vectorizing mathematical functions
10.3. Branch-free sorting
11. Pipelining
12. Caches
12.3. Efficient Cholesky factorization
13. Parallelization
ODEs
14. Solving ordinary differential equations
15. A little bit of theory
16. Some simple time-stepping methods
17. Implementation
17.1. Function algebra
17.2. Implementing the explicit Euler method
17.3. Implementing a Newton solver
17.4. Coding the Implicit Euler method
18. Automatic Differentiation
19. Runge-Kutta methods
20. Automatic step-size control
21. Mechanical Systems
PDEs
22. Partial differential equations
23. The Poisson Equation
23.1. Solving the Poisson Equation
23.2. Boundary Conditions
23.3. Approximation of functions
23.4. The electric field in a capacitor
23.5. Iterative Solvers
24. Time-dependent equations
24.1. Heat Equation
24.2. Various methods for the Heat Equation
24.3. Wave Equation
24.4. Verlet time-stepping and Mass-lumping
24.5. Waveguides
25. Elasticity
25.1. Modeling Elasticity
25.2. Newton’s method
25.3. Solving nonlinear Elasticity
25.4. 3D Solid Mechanics
25.5. Elastodynamics with Newmark time-stepping
25.6. Forces and Moments
25.7. Meta-material with negative
\(\nu\)
25.8. Exercises
26. Navier Stokes Equations:
26.1. Stokes Equation
26.2. Instationary transport equation
26.3. Benchmark: Flow around a cylinder
26.4. NACA airfoil
26.5. Tesla Valve
27. Maxwell’s equations
27.1. Magnetostics
27.2. A simple coil
27.3. An advanced coil
27.4. Exercises
27.5. The function space H(curl)
27.6. The de Rham complex
27.7. Ranks of discrete operators
28. Helmholtz Equation
29. Eigenvalue problems
30. Stationary Transport Equation
31. Instationary Transport Equation
32. Navier Stokes Equations
33. Computation of Curvature
FMM
34. Fast Multipole Methods
35. Multipole basis functions
36. Layer potentials
Index
