arXiv subjects
Publications and source records attributed to Jean-Luc Guermond.
A non-intrusive, discretization and PDE agnostic limiting technique is introduced. The method is conservative and can be applied to high-order approximations including spectral methods. It can be applied to time-dependent and time-independent problems. It can be used for elliptic, parabolic and hyperbolic problems.
In this work we introduce an implicit-explicit invariant-domain preserving approximation of the nonequilibrium gray radiation-hydrodynamics equations. A time and space approximation of the system is proposed using a novel split of the equations composed of three elementary subsystems, two hyperbolic and one parabolic. The approximation thus realized is proved to be consistent, conservative, invariant-domain preserving, and first-order accurate. The proposed method is a stepping stone for achieving higher-order accuracy in space and time in the forthcoming second part of this work. The method is numerically illustrated and shown to converge as advertised. This paper is dedicated to the memory of Peter Lax.
The paper compares standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered (Schur complement, fully coupled system, with and without augmented Lagrangian). One investigates whether these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (number of degrees of freedom per time step per core per second). Numerical tests on fine unstructured meshes (68 millions degrees of freedoms) demonstrate GMRES/CG convergence rates that are independent of the mesh size and improve with the Reynolds number for most methods. Three conclusions are drawn: (1) Whether solving the pressure Schur complement or the fully coupled system does not make any significant difference in terms of throughput. (2) Although very good parallel scalability is observed for the augmented Lagrangian method, the best throughput is achieved without using the augmented Lagrangian formulation. (3) The throughput of all the methods tested in the paper are on average 25 times slower than that of traditional pressure-correction and velocity-correction methods. Hence, although all these methods are very efficient for solving steady state problems, none of them is unfortunately competitive for solving time-dependent problems.
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.
We introduce a high-order space-time approximation of the Shallow Water Equations with sources that is invariant-domain preserving (IDP) and well-balanced with respect to rest states. The employed time-stepping technique is a novel explicit Runge-Kutta (ERK) approach which is an extension of the class of ERK-IDP methods introduced by Ern and Guermond (SIAM J. Sci. Comput. 44(5), A3366--A3392, 2022) for systems of non-linear conservation equations. The resulting method is then numerically illustrated through verification and validation.
This paper is concerned with the approximation of the compressible Euler equations supplemented with an arbitrary or tabulated equation of state. The proposed approximation technique is robust, formally second-order accurate in space, invariant-domain preserving, and works for every equation of state, tabulated or analytic, provided the pressure is nonnegative. An entropy surrogate functional that grows across shocks is proposed. The numerical method is verified with novel analytical solutions and then validated with several computational benchmarks seen in the literature.
We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of fractional-order Sobolev spaces. By assuming additionally that the target field has a curl or divergence property, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using the quasi-interpolation errors with or without boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIM Math. Model. Numer. Anal., 51 (2017), pp.~1367--1385]. By using the face-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found. Comput. Math., (2021)], and exploiting the additional assumption made on the curl or the divergence of the target field, a localized upper bound on the quasi-interpolation error is derived. As an illustration, we show how to apply these results to the error analysis of the curl-curl problem associated with Maxwell's 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.
The objective of this paper is to propose a hyperbolic relaxation technique for the dispersive Serre-Green-Naghdi equations (also known as the fully non-linear Boussinesq equations) with full topography effects introduced in Green, A.E. and Naghdi, P.M. (J. Fluid Mech., 78, 237-246, 1976) and Seabra-Santos el al (J. Fluid Mec.h, 176, 117-134, 1997). This is done by revisiting a similar relaxation technique introduced in Guermond el al (J. Comput. Phys., 399, 108917, 2019) with partial topography effects. We also derive a family of analytical solutions for the one-dimensional dispersive Serre-Green-Naghdi equations that are used to verify the correctness the proposed relaxed model. The method is then numerically illustrated and validated by comparison with experimental results.
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.
We introduce a (linear) positive and asymptotic preserving method or solving the one-group radiation transport equation. The approximation in space is discretization agnostic: the space approximation can be done with continuous or discontinuous finite elements (or finite volumes, or finite differences). The method is first-order accurate in space. This type of accuracy is coherent with Godunov's theorem since the method is linear. The two key theoretical results of the paper are Theorem~4.4 and Theorem~4.8. The method is illustrated with continuous finite elements. It is observed to converge with the rate $\calO(h)$ in the $L^2$-norm on manufactured solutions, and it is $\calO(h^2)$ in the diffusion regime. Unlike other standard techniques, the proposed method does not suffer from overshoots at the interfaces of optically thin and optically thick regions.
In this paper we investigate the approximation of a diffusion model problem with contrasted diffusivity and the error analysis of various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than one. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. The error estimates are robust with respect to the diffusivity contrast. We briefly show how the analysis can be extended to the Maxwell's equations.
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.
We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than $\frac12$ and can be arbitrarily close to $0$ when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in $H_{\text{curl}}$- and $H_{\text{div}}$-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev smoothness index arbitrarily close to $0$. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.
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.
This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the best approximation error in any $L^p$-norm assuming regularity in the fractional Sobolev spaces $W^{r,p}$, where $p\in [1,\infty]$ and the smoothness index $r$ can be arbitrarily close to zero. The operator is stable in $L^1$, leaves the corresponding finite element space point-wise invariant whether homogeneous boundary conditions are imposed or not. The theory is illustrated on $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming spaces.
This paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the Riemann solution for the Euler system of gas dynamics with the co-volume equation of state. The novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy. The convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co-volume equation of state and the heat capacity ratio $γ$ in the range $(1,5/3]$
A conservative invariant domain preserving Arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. The method is explicit in time, works with continuous finite elements and is first-order accurate in space. One originality of the present work is that the artificial viscosity is unambiguously defined irrespective of the mesh geometry/anisotropy and does not depend on any ad hoc parameter. The proposed method is meant to be a stepping stone for the construction of higher-order methods in space by using appropriate limitation techniques.