arXiv subjects

Rémi Abgrall

Publications and source records attributed to Rémi Abgrall.

45 records · Page 3Linked to original sources

arXiv2020

General polytopal H(div) conformal finite elements and their discretisation spaces

We present a class of discretisation spaces and H(div)-conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart-Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div)-conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfill. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart-Thomas elements at each interface, for any order and any polytopial shape. Then, to close the introduction of those new elements by an example, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.

math.NA↗

arXiv2019

Neural network based limiter with transfer learning

A neural network is trained using simulation data from a Runge Kutta discontinuous Galerkin (RKDG) method and a modal high order limiter. With this methodology, we design one and two-dimensional black-box shock detection functions. Furthermore, we describe a strategy to adapt the shock detection function to different numerical schemes without the need of a full training cycle and large dataset. We evaluate the performance of the neural network on a RKDG scheme for validation. To evaluate the domain adaptation properties of this neural network limiter, our methodology is verified on a residual distribution scheme (RDS), both in one and two-dimensional problems, and on Cartesian and unstructured meshes. Lastly, we report on the quality of the numerical solutions when using a neural based shock detection method, in comparison to more traditional limiters, as well as on the computational impact of using this method in existing codes.

math.NA↗

arXiv2019

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

In the research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linear stable if the boundary conditions are done in the correct way. In this work, we extend this investigation to the non-linear case and focusing on entropy conservation. Switching to entropy variables, we will provide an estimation on the boundary operators also for non-linear problems to guarantee conservation. In numerical simulations, we verify our theoretical analysis.

math.NA↗

arXiv2019

Non-equilibrium Model for Weakly Compressible Multi-component Flows: the Hyperbolic Operator

We present a novel pressure-based method for weakly compressible multiphase flows, based on a non-equilibrium Baer and Nunziato-type model. In this work, we describe the hyperbolic operator, thus we do not consider relaxation terms. The acoustic part of the governing equations is treated implicitly to avoid the severe restriction on the time step imposed by the CFL condition at low-Mach. Particular care is taken to discretize the non-conservative terms to avoid spurious oscillations across multi-material interfaces. The absence of oscillations and the agreement with analytical or published solutions is demonstrated in simplified test cases, which confirm the validity of the proposed approach as a building block on which developing more accurate and comprehensive methods.

physics.comp-ph↗

arXiv2019

A class of finite dimensional spaces and H(div) conformal elements on general polytopes

We present a class of discretisation spaces and H(div)-conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart-Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div)-conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart-Thomas elements at each interface, for any order and any polytopial shape. Then, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.

math.NA↗

arXiv2018

Asymptotic preserving Deferred Correction Residual Distribution schemes

This work aims to extend the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve hyperbolic system of partial differential equations. Up to our knowledge, it was used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an IMEX (implicit--explicit) version of the residual distribution schemes, that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g. multiphase flows, kinetic models, viscoelasticity problems. Moreover, the provided scheme is able to catch different relaxation scales automatically, without losing accuracy. The scheme is asymptotic preserving and this guarantees that in the relaxation limit, we recast the expected macroscopic behaviour. To get a high order accuracy, we use an IMEX time discretization combined with a Deferred Correction (DeC) procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in 1D and 2D for stiff systems of equations.

math.NA↗

arXiv2018

High order residual distribution for steady state problems for hyperbolic conservation laws}

In this paper, we propose a high order residual distribution conservative finite difference scheme for solving steady state conservation laws. A new type of WENO (weighted essentially non-oscillatory) termed as WENO-ZQ integration is used to compute the numerical fluxes and source term based on the point values of the solution, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving shocks of the proposed methods.

math.NA↗

arXiv2018

Capturing near-equilibrium solutions: a comparison between high-order discontinuous Galerkin methods and well-balanced schemes

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.

math.NA↗

arXiv2015

On Third-Order Limiter Functions for Finite Volume Methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield $2^{\text{nd}}$-order accuracy, the new limiter is $3^\text{rd}$-order accurate for smooth solutions. In an earlier work, such a $3^\text{rd}$-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters.

math.NA↗