Course Objectives: The overarching theme of this course is the analysis of numerical methods for partial differential equations, and in particular time dependent problems. The course emphasizes fundamental concepts, with the primary goal being that students gain a complete understanding of how to evaluate the efficacy of numerical methods for solving various types of PDEs. This is not a survey course, although students will program methods to gain understanding of how methods work, in particular to compare advantages and disadvantages of various algorithms.

We will use linear advection and advection/diffusion in one dimension as a primary model to study important properties for numerical methods, namely accuracy, stability, convergence, and efficiency. We will then move on to non-linear conservation laws. We will mainly focus on finite difference methods, but time permitting we will introduce other algorithms such as spectral methods. We will also discuss how to handle shock discontinuities and inflow/outflow boundary conditions.

Hyperbolic partial differential equations arise in many applications, including gas dynamics, acoustics, electro-magnetics, and indeed in any field where advection transport or wave motion can be used to describe the phenomenon of interest. It can be challenging to numerically simulate hyperbolic PDEs. They are non-linear, and can induce shock formation. If not handled properly, the numerical method can yield oscillations that will eventually cause the solution to blow up. Conversely, too much dissipation can render the solution meaningless. The desire to delicately balance stability with accuracy, as well as numerical efficiency, has driven an entire field of applied mathematics. It is still a vibrant area of activity, and will increasingly need to meet new challenges as we try to handle problems of multiple scales, uncertainty, and large data sets. It is critical to have a fundamental understanding of simple problems in order to meet these new challenges.