Prerequisites: Familiarity with Taylor series, norms, orthogonal polynomials, matrix analysis, linear systems of equations, eigenvalues, differential equations, and programming in MATLAB or a similar language. The course covers theory and algorithms for the numerical solution of ODEs and of PDEs of parabolic, hyperbolic and elliptic type. Theoretical concepts include: accuracy, zero-stability, absolute stability, convergence, order of accuracy, stiffness, conservation and the CFL condition. Algorithms covered include: finite differences, steady and unsteady discretization in one and two dimensions, Newton methods, Runge-Kutta methods, linear multistep methods, multigrid, implicit methods for stiff problems, centered and upwind methods for wave equations, dimensional splitting and operator splitting.