Abstract

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space-time adaptive meshes. Two main research fields can be distinguished: (1) fully explicit DG methods on collocated grids and (2) semi-implicit DG methods on edge-based staggered grids. DG methods became increasingly popular in the last twenty years mainly because of three intriguing properties: i) non-linear L2 stability has been proven; ii) arbitrary high order of accuracy can be achieved by simply increasing the polynomial order of the chosen basis functions, used for approximating the state-variables; iii) high scalability properties make DG methods suitable for large-scale simulations on general unstructured meshes. It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ’Gibbs phenomenon’. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. Among them, a rather intriguing paradigm has been defined in the work of [71], in which the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. In the present work the main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Then, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. Moreover, handling typical multiscale problems, dynamic adaptive mesh refinement (AMR) and adaptive polynomial order methods are probably the two main ways of preserving accuracy and efficiency, and saving computational effort. The here adopted AMRapproach is the so called ’cell by cell’ refinement because of its formally very simple tree-type data structure. In the here-presented ’cell-by-cell’ AMR every single element is recursively refined, from a coarsest refinement level l0 = 0 to a prescribed finest (maximum) refinement level lmax, accordingly to a refinement-estimator function X that drives step by step the choice for recoarsening or refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. First, the Euler equations of compressible gas dynamics and the magnetohydrodynamics (MHD) equations have been treated [281]. Then, the presented method has been readily extended to the special relativistic ideal MHD equations [280], but also the the case of diffusive fluids, i.e. fluid flows in the presence of viscosity, thermal conductivity and magnetic resistivity [116]. In particular, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within the cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). The resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, from low to high Mach numbers, from low to high Reynolds regimes. In particular, concerning MHD equations, the divergence-free character of the magnetic field is taken into account through the so-called hyperbolic ’divergence-cleaning’ approach which allows to artificially transport and spread the numerical spurious ’magnetic monopoles’ out of the computational domain. A special treatment has been followed for the incompressible Navier-Stokes equations. In fact, the elliptic character of the incompressible Navier-Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand [117]. In this context, we derived two new families of spectral semi-implicit and spectral space-time DG methods for the solution of the two and three dimensional Navier-Stokes equations on edge-based staggered Cartesian grids [115], following the ideas outlined in [97] for the shallow water equations. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor theta in [0.5, 1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. Moreover, a rigorous theoretical analysis of the condition number of the resulting linear systems and the design of specific preconditioners, using the theory of matrix-valued symbols and Generalized Locally Toeplitz (GLT) algebra has been successfully carried out with promising results in terms of numerical efficiency [102]. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist. Moreover, the here mentioned semi-implicit DG method has been successfully extended to a novel edge-based staggered ’cell-by-cell’ adaptive meshes [114].