A New Ensemble HDG Method for Parameterized Convection Diffusion PDEs

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.

References in corpus (3)

• A second-order Ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier-Stokes equations
• A hybridizable discontinuous Galerkin method for computing nonlocal electromagnetic effects in three-dimensional metallic nanostructures
• A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations