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In the context of Discontinuous Galerkin Spectral Element Methods (DGSEM), τ -estimation has been successfully used for p-adaptation algorithms. This method estimates the truncation error of representations with different polynomial orders using the solution on a reference mesh of relatively high order.In this paper, we present a novel anisotropic truncation error estimator derived from the τ -estimation procedure for DGSEM. We exploit the tensor product basis properties of the numerical solution to design a method where the total truncation error is calculated as a sum of its directional components. We show that the new error estimator is cheaper to evaluate than previous implementations of the τ -estimation procedure and that it obtains more accurate extrapolations of the truncation error for representations of a higher order than the reference mesh. The robustness of the method allows performing the p-adaptation strategy with coarser reference solutions, thus further reducing the computational cost. The proposed estimator is validated using the method of manufactured solutions in a test case for the compressible Navier-Stokes equations.

High-order discontinuous Galerkin methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the order also increases the computational cost. Several techniques have been introduced in the past to reduce this cost. On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy. On the other hand, multigrid solvers may accelerate time marching computations for a fixed number of degrees of freedom.In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error estimator. To achieve this, we show that a recently developed anisotropic truncation error estimator [1, A. M. Rueda-Ramírez, G. Rubio, E. Ferrer, E. Valero, Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method, Journal of Scientific Computing] is perfectly suited to be performed inside the multigrid cycle with negligible extra cost. Furthermore, we introduce a multi-stage p-adaptation procedure which reduces the computational time when very accurate results are required.The proposed methods are tested for the compressible Navier-Stokes equations, where we investigate two cases. First, the 2D boundary layer flow on a flat plate is studied to assess accuracy and computational cost of the algorithm, where a speed-up of 816 is achieved compared to the traditional explicit method. Second, the 3D flow around a sphere is simulated and used to test the anisotropic properties of the proposed method, where a speed-up of 152 is achieved compared to the explicit method. The proposed multi-stage procedure achieved a speed-up of 2.6 in comparison to the single-stage method in highly accurate simulations.

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We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advectiondiffusion partial differential equations in conservation form. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summationby-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

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High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.