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

Truncation Error-Based Anisotropic pp-Adaptation for Unsteady Flows for High-Order Discontinuous Galerkin Methods

In this work, we extend the $\tau$-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 $p$-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 $\tau$-estimation strategy to unsteady problems. Finally, we present and compare two adaptation strategies for unsteady problems: the dynamic and static $p$-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 $p$-adaptation process is performed after several estimation stages, using a combination of the periodic error measures. The static $p$-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 $p$-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 $\times4.5$ for the Euler test case and $\times2.2$ for the Navier-Stokes test case

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

A non-adapted sparse approximation of PDEs with stochastic inputs

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems (with slow decay in the spectrum)

Divergence error based pp-adaptive discontinuous Galerkin solution of time-domain Maxwell's equations

A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive method. The nature of divergence error is explored and that it is a direct consequence of the act of discretization is established. Its relation with relative truncation error is formed which enables the use of divergence error as an inexpensive proxy to truncation error. Divergence error is used as an indicator to dynamically identify and assign spatial operators of varying accuracy to substantial regions in the computational domain. This results in a reduced computational cost than a comparable discontinuous Galerkin time-domain solution using uniform degree piecewise polynomial bases throughout.Comment: 28 pages, 22 figure

Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems

Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge-Kutta schemes available in literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.Comment: 37 pages, 3 pages of appendi