Abstract
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.
Highlights
The hybridisable discontinuous Galerkin (HDG) method, proposed by Cockburn and co-workers [1,2,3,4,5], has gained popularity in the last decade due to its ability to reduce the global number of coupled degrees of freedom required by other DG approaches
This work proposes a dual time stepping (DTS) approach to solve the global system of equations that appear in the HDG method for a convection-diffusion model problem
Numerical experiments were used to illustrate the dependence of the critical dual time step with respect to the physical and the numerical parameters
Summary
The hybridisable discontinuous Galerkin (HDG) method, proposed by Cockburn and co-workers [1,2,3,4,5], has gained popularity in the last decade due to its ability to reduce the global number of coupled degrees of freedom required by other DG approaches. As with other implicit methods, the computational cost and memory requirements of the HDG method can become prohibitive when applied to problems that require a large number of degrees of freedom This is of major importance for non-linear and/or transient problems in three dimensions [23,18,13]. The performance of the proposed DTS approach is analysed for steady and transient problems, by comparing the minimum number of dual time steps required to reach the steady state solution in each case. The proposed DTS technique for the HDG global problem is presented, including the proof of existence and uniqueness of the steady state solution, the von Neumann stability analysis of the DTS approach and the optimal choice of the dual time step. Appendix A presents a brief analysis of the stability and the optimal choice for the dual time step based on the eigenvalues of the global HDG matrix
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