eHDG:An Exponentially Convergent Iterative Solver for HDG Discretizations of Hyperbolic Partial Differential Equations

TL;DR

This paper introduces eHDG, an iterative solver for HDG discretizations of hyperbolic PDEs, which is scalable, exponentially convergent, and suitable for high-performance computing systems.

Contribution

The paper presents a novel fixed-point iterative solver for HDG methods that guarantees exponential convergence independent of solution order, enhancing computational efficiency.

Findings

Proven exponential convergence in theory and practice.

Convergence rate is independent of solution polynomial order.

Numerical results confirm theoretical predictions for 2D and 3D problems.

Abstract

We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. We rigorously show that the proposed method is exponentially convergent in the number of iterations for transport and linearized shallow water equations. Furthermore, the convergence is independent of the solution order. Various 2D and 3D numerical results for steady and time-dependent problems are presented to verify our theoretical findings.