5 research outputs found
Quasi-a priori truncation error estimation in the DGSEM
In this paper we show how to accurately perform a quasi-a priori estimation of the truncation error of steady-state solutions computed by a discontinuous Galerkin spectral element method. We estimate the spatial truncation error using the ?-estimation procedure. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, we use non time-converged solutions on one grid with different polynomial orders. The quasi-a priori approach estimates the error while the residual of the time-iterative method is not negligible. Furthermore, the method permits one to decouple the surface and the volume contributions of the truncation error, and provides information about the anisotropy of the solution as well as its rate of convergence in polynomial order. First, we focus on the analysis of one dimensional scalar conservation laws to examine the accuracy of the estimate. Then, we extend the analysis to two dimensional problems. We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the truncation error
Truncation Error-Based Anisotropic -Adaptation for Unsteady Flows for High-Order Discontinuous Galerkin Methods
In this work, we extend the -estimation method to unsteady problems and
use it to adapt the polynomial degree for high-order discontinuous Galerkin
simulations of unsteady flows. The adaptation is local and anisotropic and
allows capturing relevant unsteady flow features while enhancing the accuracy
of time evolving functionals (e.g., lift, drag). To achieve an efficient and
unsteady truncation error-based -adaptation scheme, we first revisit the
definition of the truncation error, studying the effect of the treatment of the
mass matrix arising from the temporal term. Secondly, we extend the
-estimation strategy to unsteady problems. Finally, we present and
compare two adaptation strategies for unsteady problems: the dynamic and static
-adaptation methods. In the first one (dynamic) the error is measured
periodically during a simulation and the polynomial degree is adapted
immediately after every estimation procedure. In the second one (static) the
error is also measured periodically, but only one -adaptation process is
performed after several estimation stages, using a combination of the periodic
error measures. The static -adaptation strategy is suitable for
time-periodic flows, while the dynamic one can be generalized to any flow
evolution.
We consider two test cases to evaluate the efficiency of the proposed
-adaptation strategies. The first one considers the compressible Euler
equations to simulate the advection of a density pulse. The second one solves
the compressible Navier-Stokes equations to simulate the flow around a cylinder
at Re=100. The local and anisotropic adaptation enables significant reductions
in the number of degrees of freedom with respect to uniform refinement, leading
to speed-ups of up to for the Euler test case and for
the Navier-Stokes test case
Adaptation strategies for high order discontinuous Galerkin methods based on Tau-estimation
In this paper three p-adaptation strategies based on the minimization of the truncation error are presented for high order discontinuous Galerkin methods. The truncation error is approximated by means of a ? -estimation procedure and enables the identification of mesh regions that require adaptation. Three adaptation strategies are developed and termed a posteriori, quasi-a priori and quasi-a priori corrected. All strategies require fine solutions, which are obtained by enriching the polynomial order, but while the former needs time converged solutions, the last two rely on non-converged solutions, which lead to faster computations. In addition, the high order method permits the spatial decoupling for the estimated errors and enables anisotropic p-adaptation. These strategies are verified and compared in terms of accuracy and computational cost for the Euler and the compressible Navier?Stokes equations. It is shown that the two quasi- a priori methods achieve a significant reduction in computational cost when compared to a uniform polynomial enrichment. Namely, for a viscous boundary layer flow, we obtain a speedup of 6.6 and 7.6 for the quasi-a priori and quasi-a priori corrected approaches, respectively