Theory of linear and nonlinear hyperbolic PDEs, with applications including fluid dynamics, elasticity, acoustics, electromagnetics, shallow water waves, and traffic modeling. Theory of shock and rarefaction waves. Finite volume methods for numerical approximation of solutions; Godunov's method, TVD methods, and high order methods. Stability, convergence, and entropy conditions. Numerical solution of multidimensional problems. The course covers theory and algorithms for the numerical solution of linear and nonlinear hyperbolic PDEs, with applications including fluid dynamics, elasticity, acoustics, electromagnetics, shallow water waves, and traffic flow. The main concepts include: characteristics; shock and rarefaction waves; weak solutions; entropy; the Riemann problem; finite volume methods; Godunov’s method; TVD methods; and high order methods; stability, accuracy, and convergence of numerical solutions.