A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

Abstract

SUMMARY We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.

Anisotropic hp-adaptive discontinuous Galerkin method for the numerical solution of time dependent PDEs

“…For their form, see, e.g., [37] and for their DG discretization, e.g., [38,34]. These problems are more complicated since they can involve several physical features.…”

“…Similarly as in [39][40][41]34], we consider the viscous interaction of a plane weak shock wave with a single isentropic vortex. During the interaction, acoustic waves are produced, and we investigate the ability of the numerical scheme to capture these waves.…”

“…The Jacobian matrix is replaced by the so-called flux matrix (see [34]) which can be derived analytically. The Newton-like iterative process is performed until the algebraic error does not significantly influence the space discretization error.…”

Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows

“…This estimator is often split into its spatial and temporal parts which reflect the space and time discretization separately (in some sense). In [13], we presented a different approach, where the spatial error is considered as a difference between the approximate (=space-time discrete) solution and the time semi-discrete solution (which is formally exact with respect to space). Similarly, the temporal error is considered as a difference between the approximate solution and the space semi-discrete solution (which is formally exact with respect to time).…”

“…This approach has the advantage of permitting to use large time steps. In [13], we employed high order multistep backward difference formulae (BDF) for the time discretization.…”