TIME: 3:30 PM

LOCATION: GMCS 314

SPEAKER: Mohammad Farazmand, North Carolina State University

ABSTRACT: Classical spectral methods for numerically solving PDEs assume the solution as the linear combination of prescribed basis functions (or modes). This assumption limits their efficacy for time-dependent multi-scale problems with localized time-varying features. I will first introduce the notion of Shape-Morphing Solutions (SMS), which allow the modes to change shape over time, in a predictable fashion, in order to automatically adapt to the true solution of the PDE. This allows the SMS to approximate the PDE’s solution more economically. I will also discuss several attractive features of SMS: (i) It is also easy to enforce known conserved quantities of the PDE in the numerical solution. (ii) Classical Galerkin methods turn out to be a special case of SMS. (iii) Interpretation of SMS as a neural network with time-dependent weights and biases. (iv) Efficient reduced-order modeling with SMS. Finally, I demonstrate the application of SMS to several problems including rogue waves, vortex dynamics, and heat transfer.

HOST: Efstathis Charalampidis