Madison Phelps:

Nonlinear Solvers for Permafrost Problems

Abstract: We study the nonlinear heat equation with phase change between liquid and ice with the enthalpy-temperature relationship, w-theta, in three variants. For Stefan problem (ST), this relationship is usually a multivalued graph where the phase transition resembles a shifted Heaviside function. For permafrost problems (P), w-theta is continuous, but its inverse requires a nonlinear solver. For soil with trapped air and large pores, we consider a model denoted (P*) where w-theta includes a multivalued graph but also has piecewise continuous properties. When the model is discretized in space by FV (finite volumes) and in time (implicitly), we require nonlinear solvers at two levels: a global one for the PDE, and a local one at each grid cell of the PDE. We give an overview of the choices available solvers including the recently popular Newton-Anderson variants which we test on simple model problems. Then we show their performance for the (ST, P, and P*) models and illustrate with numerical simulations.

Praveeni Mathangadeera:

Computational modeling of the nonlinear heat equation in frozen soil and snow.

Abstract: Heat conduction in snow is modeled by a nonlinear parabolic partial differential equation which modifies the Stefan problem of liquid/ice phase transition with a free boundary. In soil, heat conduction is a nonlinear parabolic equation which is smoother than the Stefan problem. We discuss and compare three time stepping formulations for discretization of these models: the fully implicit, sequential and semi-implicit approaches. For the soil problem we also compare the use of different primary unknowns. Next, we explore the pointwise snow model which solves a single algebraic equation instead of a PDE and thus approximates the snow PDE model. This pointwise model involves n=11 environmental parameters including the albedo of surface. We analyze the sensitivity of the solutions and set up a ML regression model: that allow us to assess the robustness of our computational model as well as to understand the uncertainty associated with the parameters and the model itself. Our simulations and analyses contribute to the understanding of the response of the soils in the Arctic to the weather data and help to assess the reliability of the model.