arXiv subjects

Alessia Lucca

Publications and source records attributed to Alessia Lucca.

3 recordsLinked to original sources

arXiv2025

Structure-preserving schemes for nonlinear symmetric hyperbolic and thermodynamically compatible systems of partial differential equations

This paper aims at developing exactly energy-conservative and structure-preserving finite volume schemes for the discretisation of first-order symmetric-hyperbolic and thermodynamically compatible (SHTC) systems of partial differential equations in continuum physics. Due to their thermodynamic compatibility the class of SHTC systems satisfies an additional conservation law for the total energy and many PDE in this class of equations also satisfy stationary differential constraints (involutions). First, we propose a simple semi-discrete cell-centered HTC finite volume scheme that employs collocated grids and that is compatible with the total energy conservation law, but which does not satisfy the involutions. Second, we develop a fully discrete semi-implicit finite volume scheme that conserves total energy and which can be proven to satisfy also the involution constraints exactly at the discrete level. This method is a vertex-based staggered semi-implicit scheme that preserves the basic vector calculus identities $\nabla \cdot \nabla \times A = 0$ and $\nabla \times \nabla \phi = 0$ for any vector and scalar field, respectively, exactly at the discrete level and which is also exactly totally energy conservative. The main key ingredient of the proposed implicit scheme is the fact that it uses a discrete version of the symmetric-hyperbolic Godunov-form of the governing PDE system. This leads naturally to sequences of symmetric and positive definite linear algebraic systems to be solved inside an iterative fixed-point method used in each time step. We apply our new schemes to three different SHTC systems. In particular, we consider the equations of nonlinear acoustics, the nonlinear Maxwell equations in the absence of charges and a nonlinear version of the Maxwell-GLM system. We also show some numerical results to provide evidence of the stated properties of the proposed schemes.

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arXiv2024

Variational derivation and compatible discretizations of the Maxwell-GLM system

We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes in order to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general-purpose and non-structure-preserving numerical schemes for hyperbolic PDE. Among the many mathematically interesting features of the model are: i) its symmetric hyperbolicity, ii) the extra conservation law for the total energy density and, most importantly, iii) the very peculiar combination of the basic differential operators, since both, curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and asymptotic-preserving finite volume schemes for its discretization.

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arXiv2023

An implicit staggered hybrid finite volume/finite element solver for the incompressible Navier-Stokes equations

We present a novel fully implicit hybrid finite volume/finite element method for incompressible flows. Following previous works on semi-implicit hybrid FV/FE schemes, the incompressible Navier-Stokes equations are split into a pressure and a transport-diffusion subsystem. The first of them can be seen as a Poisson type problem and is thus solved efficiently using classical continuous Lagrange finite elements. On the other hand, finite volume methods are employed to solve the convective subsystem, in combination with Crouzeix-Raviart finite elements for the discretization of the viscous stress tensor. For some applications, the related CFL condition, even if depending only in the bulk velocity, may yield a severe time restriction in case explicit schemes are used. To overcome this issue an implicit approach is proposed. The system obtained from the implicit discretization of the transport-diffusion operator is solved using an inexact Newton-Krylov method, based either on the BiCStab or the GMRES algorithm. To improve the convergence properties of the linear solver a symmetric Gauss-Seidel (SGS) preconditioner is employed, together with a simple but efficient approach for the reordering of the grid elements that is compatible with MPI parallelization. Besides, considering the Ducros flux for the nonlinear convective terms we can prove that the discrete advection scheme is kinetic energy stable. The methodology is carefully assessed through a set of classical benchmarks for fluid mechanics. A last test shows the potential applicability of the method in the context of blood flow simulation in realistic vessel geometries.

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