Course on Numerical Methods for PDEs

Numerical Methods for PDEs

Professor: Gustavo Buscaglia.; e-mail: gustavo.buscaglia at icmc.usp.br

Summary:: This is a classical graduate course on this subject. The emphasis is put on finite volume methods. The first topic is a revision of truncation errors, Taylor expansions, discrete approximations, definition of convergence, and finite difference formulae.; The second topic is elliptic equations. Existence and uniqueness results. Finite difference discretization. Finite volume discretization for discontinuous coefficients. Two and three dimensional problems. Conservation at the interface. Convergence. Nonlinear coefficients. Newton scheme.; The third topic is parabolic equations. Existence and uniqueness results. Stability. Von Neumann analysis. Lax's theorem. Implicit and explicit methods. Alternating direction and fractional step methods.; The fourth topic is hyperbolic equations. Existence and uniqueness results in the linear case. Characteristic lines. Propagation of information. Conservative and advective forms of the equation. Basic finite volume scheme. Lax-Friedrichs and Lax-Wendroff methods. Upwind method. The CFL condition. Godunov's lemma. Non-linear equations. Burgers and traffic-flow equations. Spontaneous generation of discontinuities. Rankine-Hugoniot conditions. Weak form. Shock waves and rarefaction fans. Lax-Wendroff theorem. Viscosity solutions. Artificial diffusion. Higher-order methods.; Notice that nonlinear hyperbolic systems are not discussed.; Text: Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque (SIAM), the book on CFD by P. Wesseling, and the chapter on evolution problems of the book by Dautray and Lions.