Numerical Analysis(D)
NUMERICAL ANALYSIS (D)
Part IB Numerical Analysis is essential and Analysis II, Linear Algebra and Complex Methods or Complex Analysis are all desirable.
Finite difference methods for the Poisson’s equation
- Approximation of ∇2 by finite differences. The accuracy of the five-point method in a square. Higher order methods. Solution of the difference equations by iterative methods, including multigrid. Fast Fourier transform (FFT) techniques. [5]
Finite difference methods for initial value partial differential equations
- Difference schemes for the diffusion equation and the advection equation. Proof of convergence in simple cases. The concepts of well posedness and stability. Stability analysis by eigenvalue and Fourier techniques. Splitting methods. [6]
Spectral methods
- Brief review of Fourier expansions. Calculation of Fourier coefficients with FFT. Spectral methods for the Poisson equation in a square with periodic boundary conditions. Chebyshev polynomials and Chebyshev methods. Spectral methods for initial-value PDEs. [5]
Iterative methods for linear algebraic systems
- Iterative methods, regular splittings and their convergence. Jacobi and Gauss-Seidel methods. Krylov spaces. Conjugate gradients and preconditioning. [5]
Computation of eigenvalues and eigenvectors
- The power method and inverse iteration. Transformations to tridiagonal and upper Hessenberg forms. The QR algorithm for symmetric and general matrices, including shifts. [3]
Appropriate books
- G.H. Golub and C.F. van Loan Matrix Computations. Johns Hopkins Press 1996
- A. Iserles A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press 1996
- K.W. Morton and D.F. Mayers Numerical Solution of Partial Differential Equations: an Introduction. Cambridge University Press 2005
Associated GitHub page
https://jaircambridge.github.io/Numerical-Analysis-D/
sooo0oooys 