arXiv subjects

Francesco Fambri

Publications and source records attributed to Francesco Fambri.

15 recordsLinked to original sources

arXiv2024

Structure preserving hybrid Finite Volume Finite Element method for compressible MHD

In this manuscript we present a novel and efficient numerical method for the compressible viscous and resistive MHD equations for all Mach number regimes. The time-integration strategy is a semi-implicit splitting, combined with a hybrid finite-volume and finite-element (FE) discretization in space. The non-linear convection is solved by a robust explicit FV scheme, while the magneto-acoustic terms are treated implicitly in time. The resulting CFL stability condition depends only on the fluid velocity, and not on the Alfvénic and acoustic modes. The magneto-acoustic terms are discretized by compatible FE based on a continuous and a discrete de Rham complexes designed using Finite Element Exterior Calculus (FEEC). Thanks to the use of FEEC, energy stability, magnetic-helicity conservation and the divergence-free conditions can be preserved also at the discrete level. A very efficient splitting approach is used to separate the acoustic and the Alfvénic modes in such a fashion that the original symmetries of the PDE governing equations are preserved. In this way, the algorithm relies on the solution of linear, symmetric and positive-definite algebraic systems, that are very efficiently handled by the simple matrix-free conjugate-gradient method. The resulting algorithm showed to be robust and accurate in low and high Mach regimes even at large Courant numbers. Non-trivial tests are solved in one-, two- and three- space dimensions to confirm the robustness, accuracy, and the low-dissipative and conserving properties of the final algorithm. While the formulation of the method is very general, numerical results for a second-order accurate FV-FE scheme will be presented.

math.NA↗

arXiv2023

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier-Stokes equations

In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization which preserve these invariants at the fully discrete level and we analyse its well-posedness in terms of a CFL condition. Numerical test cases performed with spline finite elements allow us to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

math.NA↗

arXiv2021

Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations

This paper presents an arbitrary h.o. accurate ADER DG method on space-time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent PDE for compr. dissipative flows: the compr. Navier-Stokes equations and the equations of visc. and res. MHD in 2 and 3 space-dimensions. The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell FV limiting procedure suitable for DG methods. It is a well known fact that a major weakness of h.o. DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. 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. An important feature of our new scheme is its ability to cure even floating point errors that may occur during a simulation, for example when taking real roots of negative numbers or after divisions by zero. We apply the whole approach for the first time to the equations of compr. gas dynamics and MHD in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme stands out against a wide number of non-trivial test cases both for the compr. Navier-Stokes and the viscous and resistive MHD equations. The present results show clearly that the shock-capturing capability of the news schemes are significantly enhanced within a cell-by-cell Adaptive Mesh Refinement implementation together with time accurate local time stepping (LTS).

math.NA↗

arXiv2021

A novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics

In this work we introduce a novel semi-implicit structure-preserving finite-volume/finite-difference scheme for the viscous and resistive equations of magnetohydrodynamics (MHD) based on an appropriate 3-split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field and a third subsystem involving the pressure-velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfven waves and the magneto-acoustic waves are treated implicitly. The final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence-free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix-free conjugate gradient algorithm. One of the peculiarities of the presented algorithm is that the magnetic field is defined on the edges of the main grid, while the electric field is on the faces. The final scheme can be regarded as a novel shock-capturing, conservative and structure preserving semi-implicit scheme for the nonlinear viscous and resistive MHD equations. Several numerical tests are presented to show the main features of our novel solver: linear-stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a 2nd-order of convergence is numerically estimated; shock-capturing capabilities are proven against a standard set of stringent MHD shock-problems; accuracy and robustness are verified against a nontrivial set of 2- and 3-dimensional MHD problems.

math.NA↗

arXiv2020

ExaHyPE: An Engine for Parallel Dynamically Adaptive Simulations of Wave Problems

ExaHyPE ("An Exascale Hyperbolic PDE Engine") is a software engine for solving systems of first-order hyperbolic partial differential equations (PDEs). Hyperbolic PDEs are typically derived from the conservation laws of physics and are useful in a wide range of application areas. Applications powered by ExaHyPE can be run on a student's laptop, but are also able to exploit thousands of processor cores on state-of-the-art supercomputers. The engine is able to dynamically increase the accuracy of the simulation using adaptive mesh refinement where required. Due to the robustness and shock capturing abilities of ExaHyPE's numerical methods, users of the engine can simulate linear and non-linear hyperbolic PDEs with very high accuracy. Users can tailor the engine to their particular PDE by specifying evolved quantities, fluxes, and source terms. A complete simulation code for a new hyperbolic PDE can often be realised within a few hours - a task that, traditionally, can take weeks, months, often years for researchers starting from scratch. In this paper, we showcase ExaHyPE's workflow and capabilities through real-world scenarios from our two main application areas: seismology and astrophysics.

cs.MS↗

arXiv2019

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions. The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwell-type evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints. The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.

math.NA↗

arXiv2019

A new causal general relativistic formulation for dissipative continuum fluid and solid mechanics and its solution with high-order ADER schemes

We present a unified causal general relativistic formulation of dissipative and non-dissipative continuum mechanics. The presented theory is the first general relativistic theory that can deal simultaneously with viscous fluids as well as irreversible deformations in solids and hence it also provides a fully covariant formulation of the Newtonian continuum mechanics in arbitrary curvilinear spacetimes. In such a formulation, the matter is considered as a Riemann-Cartan manifold with non-vanishing torsion and the main field of the theory being the non-holonomic basis tetrad field also called four-distortion field. Thanks to the variational nature of the governing equations, the theory is compatible with the variational structure of the Einstein field equations. Symmetric hyperbolic equations are the only admissible equations in our unified theory and thus, all perturbations propagate at finite speeds (even in the diffusive regime) and the Cauchy problem for the governing PDEs is locally well-posed for arbitrary and regular initial data which is very important for the numerical treatment of the presented model. Nevertheless, the numerical solution of the discussed hyperbolic equations is a challenging task because of the presence of the stiff algebraic source terms of relaxation type and non-conservative differential terms. Our numerical strategy is thus based on an advanced family of high-accuracy ADER Discontinuous Galerkin and Finite Volume methods which provides a very efficient framework for general relaxation hyperbolic PDE systems. An extensive range of numerical examples is presented demonstrating the applicability of our theory to relativistic flows of viscous fluids and deformation of solids in Minkowski and curved spacetimes.

gr-qc↗

arXiv2018

Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine

In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication avoiding and cache blocking and are furthermore very well-suited for vectorization, so that they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing level of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.

math.NA↗

arXiv2018

An efficient semi-implicit method for three-dimensional non-hydrostatic flows in compliant arterial vessels

Blood flow in arterial systems can be described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the elasticity of the arterial walls. In this article blood is treated as an incompressible Newtonian fluid that flows through compliant vessels of general cross section. A three-dimensional semi-implicit finite difference and finite volume model is derived so that numerical stability is obtained at a low computational cost on a staggered grid. The key idea of the method consists in a splitting of the pressure into a hydrostatic and a non-hydrostatic part, where first a small quasi-one-dimensional nonlinear system is solved for the hydrostatic pressure and only in a second step the fully three-dimensional non-hydrostatic pressure is computed from a three-dimensional nonlinear system as a correction to the hydrostatic one. The resulting algorithm is robust, efficient, locally and globally mass conservative and applies to hydrostatic and non hydrostatic flows in one, two and three space dimensions. These features are illustrated on nontrivial test cases for flows in tubes with circular or elliptical cross section where the exact analytical solution is known. Test cases of steady and pulsatile flows in uniformly curved rigid and elastic tubes are presented. Wherever possible, axial velocity development and secondary flows are shown and compared with previously published results.

math.NA↗

arXiv2018

ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static, i.e., we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully-discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local timestepping. In order to deal with shock waves and other discontinuities, the highorder DG schemes are supplemented with a novel a-posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed spacetimes. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes.

physics.comp-ph↗

arXiv2018

A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics

In this paper we present a novel pressure-based semi-implicit finite volume solver for the equations of compressible ideal, viscous and resistive magnetohydrodynamics (MHD). The new method is conservative for mass, momentum and total energy and in multiple space dimensions it is constructed in such a way as to respect the divergence-free condition of the magnetic field exactly, also in the presence of resistive effects. This is possible via the use of multi-dimensional Riemann solvers on an appropriately staggered grid for the time evolution of the magnetic field and a double curl formulation of the resistive terms. The new semi-implicit method for the MHD equations proposed here discretizes all terms related to the pressure in the momentum equation and the total energy equation implicitly, making again use of a properly staggered grid for pressure and velocity. The time step of the scheme is restricted by a CFL condition based only on the fluid velocity and the Alfv\'en wave speed and is not based on the speed of the magnetosonic waves. Our new method is particularly well-suited for low Mach number flows and for the incompressible limit of the MHD equations, for which it is well-known that explicit density-based Godunov-type finite volume solvers become increasingly inefficient and inaccurate due to the increasingly stringent CFL condition and the wrong scaling of the numerical viscosity in the incompressible limit. We show a relevant MHD test problem in the low Mach number regime where the new semi-implicit algorithm is a factor of 50 faster than a traditional explicit finite volume method, which is a very significant gain in terms of computational efficiency. However, our numerical results confirm that our new method performs well also for classical MHD test cases with strong shocks. In this sense our new scheme is a true all Mach number flow solver.

math.NA↗

arXiv2016

Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier-Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space-dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier-Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. The new space-time adaptive staggered DG scheme has been thoroughly verified for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier-Stokes equations on space-time adaptive meshes in two and three space dimensions.

math.NA↗

arXiv2016

Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids

In this paper two new families of arbitrary high order accurate spectral DG finite element methods are derived on staggered Cartesian grids for the solution of the inc.NS equations in two and three space dimensions. 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 a spatially staggered mesh. In the first family, h.o. of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived for the pressure gradient in the momentum equation. The resulting linear system for the pressure is symmetric and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D) and can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the NS equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly. The second family of staggered DG schemes achieves h.o. of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polynomials. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterative Picard procedure is introduced. In this manner, the symmetry and positive definiteness of the pressure system are not compromised. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary h.o. 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.

math.NA↗

arXiv2015

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

We present a new numerical tool for solving the special relativistic ideal MHD equations that is based on the combination of the following three key features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows for an arbitrary order of accuracy in both space and time, (ii) an a posteriori subcell finite volume limiter that is activated to avoid spurious oscillations at discontinuities without destroying the natural subcell resolution capabilities of the DG finite element framework and finally (iii) a space-time adaptive mesh refinement (AMR) framework with time-accurate local time-stepping. The divergence-free character of the magnetic field is instead taken into account through the so-called "divergence-cleaning" approach. The convergence of the new scheme is verified up to 5th order in space and time and the results for a set of significant numerical tests including shock tube problems, the RMHD rotor and blast wave problems, as well as the Orszag-Tang vortex system are shown. We also consider a simple case of the relativistic Kelvin-Helmholtz instability with a magnetic field, emphasizing the potential of the new method for studying turbulent RMHD flows. We discuss the advantages of our new approach when the equations of relativistic MHD need to be solved with high accuracy within various astrophysical systems.

astro-ph.HE↗

arXiv2014

Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in \cite{Dumbser2014}, the discrete solution within the troubled cells is \textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of 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. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.

math.NA↗