Wavelets and Partial Differential Equations.- What is a Wavelet?.- The Fundamental Property of Wavelets.- Wavelets for Partial Differential Equations.- High-Order Shock-Capturing Schemes for Balance Laws.- Upwind Scheme for Systems.- The Numerical Flux Function.- Nonlinear Reconstruction and High-Order Schemes.- Central Schemes.- Systems with Stiff Source.- Discontinuous Galerkin Methods: General Approach and Stability.- Time Discretization.- Discontinuous Galerkin Method for Conservation Laws.- Discontinuous Galerkin Method for Convection-Diffusion Equations.- Discontinuous Galerkin Method for PDEs Containing Higher-Order Spatial Derivatives.