Iterative methods for linear systems: Classical iterative methods (Jacobi, Gauss-Seidel and successive over relaxation (SOR) methods), Krylov subspace methods GMRES, Conjugate-gradient, biconjugate-gradient (BiCG), BiCGStab methods, preconditioning techniques, parallel implementations.
Finite difference method: Explicit and implicit schemes, consistence, stability and convergence, Lax equivalence theorem, numerical solutions to elliptic, parabolic and hyperbolic partial differential equations.

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences.

Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element methods.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford Applied Mathematics and Computing Science Series

David S. Watkins, Fundamentals of Matrix Computations, Wiley

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004.

K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press , 2nd edition (11 April 2005)