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We present a novel semi-implicit hybrid finite volume/finite element (FV/FE) method for the equations of viscous and resistive incompressible magnetohydrodynamics (MHD). The scheme preserves the divergence-free property of the magnetic field exactly on the discrete level, is second order accurate in space, and is stable in the limit of vanishing viscosity and resistivity. In particular, the MHD system is first split into a fluid and a magnetic subsystem. The former is then addressed via a semi-implicit hybrid FV/FE method. The latter belongs to a class of generalized advection-diffusion problems, for which we propose a novel semi-implicit scheme based on compatible finite element spaces and interpolation-contraction operators. In the case of vanishing resistivity, the new algorithm avoids the solution of a linear system at each time step, thanks to the framework of physical degrees of freedom, which allow to express the exterior derivative as the incidence matrix of an appropriate refinement of the original mesh. Regarding the timestepping, a local fully-discrete Lax-Wendroff-type evolution is proposed as an alternative to the classical semi-discrete method of lines. In order to suppress spurious oscillations in the presence of discontinuities or steep gradients in the magnetic field, a new simple but efficient a posteriori limiting procedure based on a nonlinear artificial resistivity is introduced. The novel methodology is validated with several test cases in two and three space dimensions using unstructured simplex meshes. [ABSTRACT FROM AUTHOR]

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time. [ABSTRACT FROM AUTHOR]

In this paper we propose a new high order accurate space–time discontinuous Galerkin (DG) finite element scheme for the solution of the linear elastic wave equations in first order velocity-stress formulation in two and three-space dimensions on staggered unstructured triangular and tetrahedral meshes. The method reaches arbitrary high order of accuracy in both space and time via the use of space–time basis and test functions. Within the staggered mesh formulation, we define the discrete velocity field in the control volumes of a primary mesh, while the discrete stress tensor is defined on a face-based staggered dual mesh. The space–time DG formulation leads to an implicit scheme that requires the solution of a linear system for the unknown degrees of freedom at the new time level. The number of unknowns is reduced at the aid of the Schur complement, so that in the end only a linear system for the degrees of freedom of the velocity field needs to be solved, rather than a system that involves both stress and velocity. Thanks to the use of a spatially staggered mesh, the stencil of the final velocity system involves only the element and its direct neighbors and the linear system can be efficiently solved via matrix-free iterative methods. Despite the necessity to solve a linear system, the numerical scheme is still computationally efficient. The chosen discretization and the linear nature of the governing PDE system lead to an unconditionally stable scheme, which allows large time steps even for low quality meshes that contain so-called sliver elements. The fully discrete staggered space–time DG method is proven to be energy stable for any order of accuracy, for any mesh and for any time step size. For the particular case of a simple Crank–Nicolson time discretization and homogeneous material, the final velocity system can be proven to be symmetric and positive definite and in this case the scheme is also exactly energy preserving . The new scheme is applied to several test problems in two and three space dimensions, providing also a comparison with high order explicit ADER-DG schemes. [ABSTRACT FROM AUTHOR]

We propose a new arbitrary high order accurate semi-implicit space–time discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and Navier–Stokes equations on staggered unstructured curved meshes. The method is pressure-based and semi-implicit and is able to deal with all Mach number flows. The new DG scheme extends the seminal ideas outlined in [1] , where a second order semi-implicit finite volume method for the solution of the compressible Navier–Stokes equations with a general equation of state was introduced on staggered Cartesian grids. Regarding the high order extension we follow [2] , where a staggered space–time DG scheme for the incompressible Navier–Stokes equations was presented. In our scheme, the discrete pressure is defined on the primal grid, while the discrete velocity field and the density are defined on a face-based staggered dual grid. Then, the mass conservation equation, as well as the nonlinear convective terms in the momentum equation and the transport of kinetic energy in the energy equation are discretized explicitly, while the pressure terms appearing in the momentum and energy equation are discretized implicitly. Formal substitution of the discrete momentum equation into the total energy conservation equation yields a linear system for only one unknown, namely the scalar pressure. Here the equation of state is assumed linear with respect to the pressure. The enthalpy and the kinetic energy are taken explicitly and are then updated using a simple Picard procedure. Thanks to the use of a staggered grid, the final pressure system is a very sparse block five-point system for three dimensional problems and it is a block four-point system in the two dimensional case. Furthermore, for high order in space and piecewise constant polynomials in time, the system is observed to be symmetric and positive definite. This allows to use fast linear solvers such as the conjugate gradient (CG) method. In addition, all the volume and surface integrals needed by the scheme depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessing stage. This leads to significant savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The viscous terms and the heat flux are also discretized making use of the staggered grid by defining the viscous stress tensor and the heat flux vector on the dual grid, which corresponds to the use of a lifting operator, but on the dual grid. The time step of our new numerical method is limited by a CFL condition based only on the fluid velocity and not on the sound speed. This makes the method particularly interesting for low Mach number flows. Finally, a very simple combination of artificial viscosity and the a posteriori MOOD technique allows to deal with shock waves and thus permits also to simulate high Mach number flows. We show computational results for a large set of two and three-dimensional benchmark problems, including both low and high Mach number flows and using polynomial approximation degrees up to p = 4 . [ABSTRACT FROM AUTHOR]

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the three-dimensional incompressible Navier–Stokes equations on staggered unstructured curved tetrahedral meshes. As is typical for space–time DG schemes, the discrete solution is represented in terms of space–time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier–Stokes equations. Similarly to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. While staggered meshes are state of the art in classical finite difference schemes for the incompressible Navier–Stokes equations, their use in high order DG schemes is still quite rare. A very simple and efficient Picard iteration is used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner, even up to very high polynomial degrees. For a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point, the resulting system is, in addition, symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data. To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the three-dimensional incompressible Navier–Stokes equations on staggered unstructured tetrahedral grids. [ABSTRACT FROM AUTHOR]