Boris D. Andrews

Boris D. Andrews, Pablo Brubeck, Patrick E. Farrell et al.

As the study of temporal and spatial discretization schemes continues to advance, recent work has focused on the use of Galerkin-in-time discretization schemes that enable broader structure-preservation than is known for Runge-Kutta integrators. While the promise of such discretizations is immense, their realization has, until now, generally relied on bespoke implementations that have limited their wider use. In this work, we present automation in Irksome for both discontinuous Galerkin and continuous Petrov-Galerkin time stepping of semidiscrete variational problems. The implementation supports auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers, and it enables switching between Runge-Kutta and Galerkin-in-time formulations with minimal changes to user code. Numerical examples illustrate accuracy, solver performance, and structure preservation across representative PDE systems.

Ganghui Zhang, Boris D. Andrews, Patrick E. Farrell

Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve these properties on discretization. This has motivated a substantial body of research on structure-preserving discretizations for these flows, albeit at low order in time. In this work, we present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov--Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. We demonstrate its structure-preserving properties and high-order convergence on several benchmark examples.