An implicit HDG method for linear convection-diffusion with dual time stepping

• Published in: Journal of computational physics 2021-06, Vol.434, p.110201, Article 110201
• Main Author: Sevilla, Ruben
• Format: Article
• Language: English
• Subjects: Computational physics; Convection; Convection-diffusion equation; Damping; Diffusion; Discontinuous Galerkin; Dual time; Hybrid method; Time marching
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110201

•Now dual time stepping approach for the HDG formulation of convection-diffusion problems.•Proof of existence of steady-state solutions in dual time.•Proof of critical an optimal dual time steps.•Numerical examples demonstrate the performance of the method.•Fact convergence achieved of convection dominated problems. This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a von Neumann analysis, leading to a closed form expression for the critical dual time step. An optimal choice for the dual time step, producing the maximum damping for all the frequencies, is also derived. Steady and transient convection-diffusion problems are considered to demonstrate the performance of the proposed DTS approach, with particular emphasis on convection dominated problems. Two simple approaches to accelerate the convergence of the DTS approach are also considered and three different time marching approaches for the dual time are compared.