A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

TL;DR

This paper introduces a second order ensemble HDG method for parameterized convection diffusion PDEs, achieving superconvergence without postprocessing, applicable to general polyhedral meshes and time-dependent coefficients.

Contribution

It develops a second order ensemble HDG method that works on arbitrary polyhedral meshes with time-dependent coefficients, improving convergence and removing the need for postprocessing.

Findings

Superconvergent rate achieved in $L^,T;L^2()$ norm.

Method applicable to general polyhedral meshes.

Numerical experiments confirm theoretical convergence rates.

Abstract

We devised a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method for a group of parameterized convection diffusion PDEs with different initial conditions, body forces, boundary conditions and coefficients in our earlier work [3]. We obtained an optimal convergence rate for the ensemble solutions in $L^\infty(0,T;L^2(\Omega))$ on a simplex mesh; and obtained a superconvergent rate for the ensemble solutions in $L^2(0,T;L^2(\Omega))$ after an element-by-element postprocessing if polynomials degree $k\ge 1$ and the coefficients of the PDEs are independent of time. In this work, we propose a new second order time stepping ensemble HDG method to revisit the problem. We obtain a superconvergent rate for the ensemble solutions in $L^\infty(0,T;L^2(\Omega))$ without an element-by-element postprocessing for all polynomials degree $k\ge 0$. Furthermore, our mesh can be any polyhedron, no need to be simplex; and the coefficients of the PDEs can dependent on time. Finally, we present numerical experiments to confirm our theoretical results.