Pierson T. Guthrey

Pierson T. Guthrey, James A. Rossmanith

Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes, such as strong stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. In particular, the maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work we introduce a novel approach that we have dubbed the regionally implicit discontinuous Galerkin (RIDG) method to overcome these small time-step restrictions. The RIDG method is based on an extension of the Lax-Wendroff DG (LxW-DG) method, which previously had been shown to be equivalent to a predictor-corrector approach, where the predictor is a locally implicit spacetime method (i.e., the predictor is something like a block-Jacobi update for a fully implicit spacetime DG method). The corrector is an explicit method that uses the spacetime reconstructed solution from the predictor step. In this work we modify the predictor to include not just local information, but also neighboring information. With this modification we show that the stability is greatly enhanced; in particular, we show that we are able to remove the polynomial degree dependence of the maximum time-step and show how this extends to multiple spatial dimensions. A semi-analytic von Neumann analysis is presented to theoretically justify the stability claims. Convergence and efficiency studies for linear and nonlinear problems in multiple dimensions are accomplished using a MATLAB code that can be freely downloaded.

William A. Sands, Pierson T. Guthrey, Nathan V. Roberts

We develop multilevel adaptive-rank iterative methods for the solution of linear and nonlinear systems arising from high-dimensional partial differential equations. Our contributions are threefold. First, we extend the projection method of Ballani and Grasedyck [6] to enable flexible preconditioning of high-dimensional linear systems in low-rank tensor formats. Second, we construct multilevel preconditioning strategies by adapting geometric multigrid methods to the low-rank setting. In contrast to prior work, which primarily employs multigrid as a standalone solver, we emphasize its role as an efficient and robust preconditioner. Third, we integrate these techniques within an inexact Newton framework for the solution of nonlinear systems. The proposed methods are evaluated on a range of model problems, including both linear and nonlinear equations, to assess their convergence behavior and computational efficiency. The results demonstrate that multilevel adaptive-rank strategies yield robust and scalable preconditioners, providing effective solvers for high-dimensional problems in low-rank formats.