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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. Mechanical Systems
PDEs
21. Partial differential equations
22. The Poisson Equation
22.1. Solving the Poisson Equation
22.2. Boundary Conditions
22.3. Approximation of functions
22.4. The electric field in a capacitor
22.5. Iterative Solvers
23. Time-dependent equations
23.1. Heat Equation
23.2. Various methods for the Heat Equation
23.3. Wave Equation
23.4. Verlet time-stepping and Mass-lumping
23.5. Waveguides
24. Elasticity
24.1. Modeling Elasticity
24.2. Newton’s method
24.3. Solving nonlinear Elasticity
24.4. 3D Solid Mechanics
24.5. Elastodynamics with Newmark time-stepping
24.6. Forces and Moments
24.7. Meta-material with negative
\(\nu\)
24.8. Exercises
25. Navier Stokes Equations:
25.1. Stokes Equation
25.2. Instationary transport equation
25.3. Benchmark: Flow around a cylinder
25.4. NACA airfoil
25.5. Tesla Valve
26. Maxwell’s equations
26.1. Magnetostics
26.2. A simple coil
26.3. An advanced coil
26.4. Exercises
26.5. The function space H(curl)
26.6. The de Rham complex
26.7. Ranks of discrete operators
27. Helmholtz Equation
28. Eigenvalue problems
29. Stationary Transport Equation
30. Instationary Transport Equation
31. Navier Stokes Equations
32. Computation of Curvature
FMM
33. Fast Multipole Methods
34. Multipole basis functions
35. Layer potentials
Index
