Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations on deforming meshes

An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady \ud ow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

An advection-robust Hybrid High-Order method for the Oseen problem

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree $\le k$ on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree $\le(k+1)$, a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity-pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element $T$ of diameter $h_T$ contributes to the discretization error with an $\mathcal{O}(h_T^{k+1})$-term in the diffusion-dominated regime, an $\mathcal{O}(h_T^{k+\frac12})$-term in the advection-dominated regime, and scales with intermediate powers of $h_T$ in between. Numerical results complete the exposition

A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward difference formulae (BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection-diffusion equations

Computational modelling of iron-ore mineralisation with stratigraphic permeability anisotropy

This study develops a computational framework to model fluid transport in sedimentary basins, targeting iron ore deposit formation. It offers a simplified flow model, accounting for geological features and permeability anisotropy as driving factors. A new finite element method lessens computational effort, facilitating robust predictions and cost-effective exploration. This methodology, applicable to other mineral commodities, enhances understanding of genetic models, supporting the search for new mineral deposits amid the global energy transition

New numerical approaches for modeling thermochemical convection in a compositionally stratified fluid

Seismic imaging of the mantle has revealed large and small scale heterogeneities in the lower mantle; specifically structures known as large low shear velocity provinces (LLSVP) below Africa and the South Pacific. Most interpretations propose that the heterogeneities are compositional in nature, differing in composition from the overlying mantle, an interpretation that would be consistent with chemical geodynamic models. Numerical modeling of persistent compositional interfaces presents challenges, even to state-of-the-art numerical methodology. For example, some numerical algorithms for advecting the compositional interface cannot maintain a sharp compositional boundary as the fluid migrates and distorts with time dependent fingering due to the numerical diffusion that has been added in order to maintain the upper and lower bounds on the composition variable and the stability of the advection method. In this work we present two new algorithms for maintaining a sharper computational boundary than the advection methods that are currently openly available to the computational mantle convection community; namely, a Discontinuous Galerkin method with a Bound Preserving limiter and a Volume-of-Fluid interface tracking algorithm. We compare these two new methods with two approaches commonly used for modeling the advection of two distinct, thermally driven, compositional fields in mantle convection problems; namely, an approach based on a high-order accurate finite element method advection algorithm that employs an artificial viscosity technique to maintain the upper and lower bounds on the composition variable as well as the stability of the advection algorithm and the advection of particles that carry a scalar quantity representing the location of each compositional field. All four of these algorithms are implemented in the open source FEM code ASPECT