arXiv subjects

Ignacio Tomas

Publications and source records attributed to Ignacio Tomas.

16 recordsLinked to original sources

arXiv2026

A structure-preserving Numerical Method for the Compressible Resistive-Hall-MHD System

In this paper, we present a structure-preserving method for the compressible resistive Hall-magnetohydrodynamics (MHD) model. The differential operator is split into two parts: a hydrodynamic part consisting of the compressible Euler equations, and a magnetic part consisting of a system coupling the Lorentz force and the induction equation. The method uses continuous Lagrange elements for the Euler part and a curl-conforming finite element space for the magnetic part. The hydrodynamic part preserves the positivity of the density and internal energy, the conservation of total energy, and the minimum principle for the specific entropy. Owing to the choice of finite elements, the magnetic part preserves the divergence involution constraint. The fluid part is solved using explicit strong-stability-preserving Runge-Kutta (SSP-RK) methods, whereas the magnetic part is solved by Crank-Nicholson method, which requires using Newton's method. Coercivity estimates for the Jacobian of the corresponding Newton iteration are presented. We introduce a high-order artificial resistivity to improve the conditioning of the nonlinear residual and the invertibility of the Jacobian. Several challenging benchmarks, including a smooth whistler wave, the Orszag-Tang vortex for comparing resistive MHD with resistive Hall-MHD, and a magnetic reconnection problem, are solved to validate the robustness and accuracy of the method.

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arXiv2025

Structure-preserving finite-element approximations of the magnetic Euler-Poisson equations

We develop a structure-preserving numerical discretization for the electrostatic Euler-Poisson equations with a given magnetic field. Our focus is on the efficient solution of problems close to the magnetic-drift limit. This regime is characterized by the co-existence of slowly moving, smooth flows with very high-frequency oscillations, spanning timescales with a difference in excess of 12 orders of magnitude. Our scheme oversteps these high-frequency oscillations with a time-step size restricted only by a hyperbolic CFL condition, while also preserving positivity of the density, positivity of the internal energy, a minimum principle for the specific entropy, and a total energy balance. The scheme uses a semi-implicit operator splitting approach composed of two subsystems: an explicit hyperbolic update with the compressible Euler equations of gas dynamics, and an implicit source update that couples the electrostatic potential, momentum, and Lorentz force. The Lorentz force is eliminated pointwise from the source system by a PDE Schur complement, so that only a single scalar, non-symmetric Poisson-like problem has to be solved per source update, which in turn enables the use of matrix-free linear algebra. Because of the efficiency of the matrix-free linear algebra, the implicit source solve costs about as much as a single explicit hyperbolic update, and accounts for less than a third of the total wall time. We illustrate the capability of the scheme by computing a diocotron instability and present growth rates that compare favorably with existing analytical results. The model, though a simplified version of the Euler-Maxwell system, represents a stepping stone toward electromagnetic solvers that are capable of working in the electrostatic and magnetic-drift limits as well as the hydrodynamic regime.

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arXiv2024

Graph-based methods for hyperbolic systems of conservation laws using discontinuous space discretizations

We present a graph-based numerical method for solving hyperbolic systems of conservation laws using discontinuous finite elements. This work fills important gaps in the theory as well as practice of graph-based schemes. In particular, four building blocks required for the implementation of flux-limited graph-based methods are developed and tested: a first-order method with mathematical guarantees of robustness; a high-order method based on the entropy viscosity technique; a procedure to compute local bounds; and a convex limiting scheme. Two important features of the current work are the fact that (i) boundary conditions are incorporated into the mathematical theory as well as the implementation of the scheme. For instance, the first-order version of the scheme satisfies pointwise entropy inequalities including boundary effects for any boundary data that is admissible; (ii) sub-cell limiting is built into the convex limiting framework. This is in contrast to the majority of the existing methodologies that consider a single limiter per cell providing no sub-cell limiting capabilities. From a practical point of view, the implementation of graph-based methods is algebraic, meaning that they operate directly on the stencil of the spatial discretization. In principle, these methods do not need to use or invoke loops on cells or faces of the mesh. Finally, we verify convergence rates on various well-known test problems with differing regularity. We propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

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arXiv2023

Structure preserving numerical methods for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wave speed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering high-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

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arXiv2023

First-order greedy invariant-domain preserving approximation for hyperbolic problems: scalar conservation laws, and p-system

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities.

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arXiv2022

Structure-preserving finite-element schemes for the Euler-Poisson equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling. The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy. A detailed discussion of algorithmic details is given, as well as proofs of the claimed properties. We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

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arXiv2022

Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties

We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.

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arXiv2022

A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations

High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes equations. Numerical experiments confirm the high order accuracy and robustness of the proposed strategy.

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arXiv2021

On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier-Stokes equations

This paper describes in detail the implementation of a finite element technique for solving the compressible Navier-Stokes equations that is provably robust and demonstrates excellent performance on modern computer hardware. The method is second-order accurate in time and space. Robustness here means that the method is proved to be invariant domain preserving under the hyperbolic CFL time step restriction, and the method delivers results that are reproducible. The proposed technique is shown to be accurate on challenging 2D and 3D realistic benchmarks.

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arXiv2020

Thermodynamically consistent physics-informed neural networks for hyperbolic systems

Physics-informed neural network architectures have emerged as a powerful tool for developing flexible PDE solvers which easily assimilate data, but face challenges related to the PDE discretization underpinning them. By instead adapting a least squares space-time control volume scheme, we circumvent issues particularly related to imposition of boundary conditions and conservation while reducing solution regularity requirements. Additionally, connections to classical finite volume methods allows application of biases toward entropy solutions and total variation diminishing properties. For inverse problems, we may impose further thermodynamic biases, allowing us to fit shock hydrodynamics models to molecular simulation of rarefied gases and metals. The resulting data-driven equations of state may be incorporated into traditional shock hydrodynamics codes.

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arXiv2020

Second-order invariant domain preserving approximation of the compressible Navier--Stokes equations

We present a fully discrete approximation technique for the compressible Navier-Stokes equations that is second-order accurate in time and space, semi-implicit, and guaranteed to be invariant domain preserving. The restriction on the time step is the standard hyperbolic CFL condition, ie $τ\lesssim \mathcal{O}(h)/V$ where $V$ is some reference velocity scale and $h$ the typical meshsize.

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arXiv2018

Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems

We introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving. The method is discretization-independent provided elementary symmetry and skew-symmetry properties are satisfied by the scheme. The method is formally first-order accurate in space. A series of higher-order methods is also introduced. When these methods violate the invariant domain properties, they are corrected by a limiting technique that we call convex limiting. After limiting, the resulting methods satisfy all the invariant domain properties that are imposed by the user (see Theorem~7.24). A key novelty is that the bounds that are enforced on the solution at each time step are necessarily satisfied by the low-order approximation.

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arXiv2017

Second-order invariant domain preserving approximation of the Euler equations using convex limiting

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.

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arXiv2016

A diffuse interface model for two-phase ferrofluid flows

We develop a model describing the behavior of two-phase ferrofluid flows using phase field-techniques and present an energy-stable numerical scheme for it. For a simplified, yet physically realistic, version of this model and the corresponding numerical scheme we prove, in addition to stability, convergence and as by-product existence of solutions. With a series of numerical experiments we illustrate the potential of these simple models and their ability to capture basic phenomenological features of ferrofluids such as the Rosensweig instability.

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arXiv2015

The equations of ferrohydrodynamics: modeling and numerical methods

We discuss the equations describing the motion of ferrofluids subject to an external magnetic field. We concentrate on the model proposed by R. Rosensweig, provide an appropriate definition for the effective magnetizing field, and explain the simplifications behind this definition. We show that this system is formally energy stable, and devise a numerical scheme that mimics the same stability estimate. We prove that solutions of the numerical scheme always exist and, under further simplifying assumptions, that the discrete solutions converge. We also discuss alternative formulations proposed in pre-existing work, primarily involving a regularization of the magnetization equation and supply boundary conditions which lead to an energy stable system. We present a series of numerical experiments which illustrate the potential of the scheme in the context of real applications.

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arXiv2013

The micropolar Navier-Stokes equations: A priori error analysis

The unsteady Micropolar Navier-Stokes Equations (MNSE) are a system of parabolic partial differential equations coupling linear velocity and pressure with angular velocity: material particles have both translational and rotational degrees of freedom. We propose and analyze a first order semi-implicit fully-discrete scheme for the MNSE, which decouples the computation of the linear and angular velocities, is unconditionally stable and delivers optimal convergence rates under assumptions analogous to those used for the Navier-Stokes equations. With the help of our scheme we explore some qualitative properties of the MNSE related to ferrofluid manipulation and pumping. Finally, we propose a second order scheme and show that it is almost unconditionally stable.

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