Instructor: Malgorzata Peszynska.
Course announcement F24 MTH 654/9 "Solving nonlinear coupled PDEs". MWF 9:00-9:50.
Instructor: Malgorzata Peszynska, mpesz@math.oregonstate.edu

Numerical approximation of coupled systems of ordinary and partial differential equations provides tools to simulate realistic scenarios (read (*) below for examples). However, construction of accurate and efficient approximation algorithms for complex models faces various interconnected mathematical and computational challenges including (i) heterogeneity of coefficients at multiple space and time scales, (ii) nonlinearity of the constitutive relationships and couplings, (iii) complexity of the resulting (nonlinear) solvers in view of the limitations of (iv) mathematical theory and of low (v) regularity of the solutions. In this class students will learn how to tackle (i-iii) while getting information on (iv-v) for selected applications. Mathematical models and theory will be introduced; algorithms and implementation will be developed, with some code templates provided. Multiphysics couplings will be handled by operator splitting or domain decomposition, and solvers will be based on recently studied variants of Newton's method e.g. semismooth or Anderson acceleration.

Roughly 1/3 of class time will be devoted to (A) the mathematical and computational background for a finite volume approximation of a coupled flow and advection-reaction-diffusion model, prototype of many (*). In the next 1/3 of class we will (B) consider gradually more complex systems; alternatively, students will be guided to tackle the multi-physics challenges specific to their own research. The remaining 1/3 of class will (C) emphasize development of realistic scenarios and presentation of simulation results of (A-B) illustrating the challenges of the underlying mathematical and computational framework.

(*) Applications modeled by a PDE/ODE or parabolic-hyperbolic PDE systems include: wildfires, flooding, sediment transport and erosion, biofilm growth in pipes, plaque build-up in arteries, freezing/thawing of ice and glacier models, traffic flow of UAVs and on networks, chemotaxis in population models, thermal treatment of tumors, neuroscience motivated electrophysiology, overland and unsaturated flow coupled to multicomponent transport with phase change under thermodynamic constraints represented by variational inequalities, radiation transport, optimal control of multi-body problem and kinematic control in robotics.

Grading will be based on Homework projects and class participation.
Preparation: students who have taken MTH 4/552-3 or similar courses on numerical PDEs should be adequately prepared. For context, MTH 654/9 in F22 focused on (iii), and MTH 654/9 in F23 focused on scalar elliptic PDEs for which (iv-v) are well understood. These - as well as other MTH 654-5-6 classes - can be taken independently from F24 MTH 654/9.
Contact: if you are interested and have questions, please contact the instructor.