Numerics for Conservation Laws
lecture notes on high-order methods for hyperbolic conservation laws
Outside of my research, I’m passionate about teaching and making high-level math concepts more accessible to a broader audience. As an undergraduate, I remember being intensely curious about numerical methods after having taken an intro numerical analysis course. The methods themselves were interesting to me but I was also drawn to the complex simulations I found in journal articles. I might not have understood the underlying methods back then, but that didn’t stop me from attempting to replicate them anyway.
Even as a Ph.D student – when I was being taught the material by some of the literal pioneers of these numerical methods – I found myself thinking how inaccessible much of this knowledge is. It’s true there are hundreds (probably thousands) of texts written on the subject, but one has to synthesize many different texts to get a clear picture of the field, many of them journal articles which may be difficult to process for undergraduates or those not in this field. And when it comes to implementation, there’s often a big gap between theory and application; the many open-source software libraries are excellent but can be a bit of a black box without the requisite background knowledge in how they work.
This project is a bit different than the others I’ve shared on this site. It doesn’t contain any novel research but rather attempts to provide a brief mathematical introduction to numerical methods for hyperbolic conservation laws, both the theory and implementation.
There are four chapters in this lecture series:
- Introduction to hyperbolic conservation laws and finite volume methods
- High-order finite volume methods and slope limiters
- WENO finite volume and finite difference methods (coming soon)
- Applications to systems of conservation laws (coming soon)
Accompanying codes in MATLAB and Python are available on my GitHub:
