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A limiting technique for scalar transport equations is presented. The originality of the method is that it does not require solving nonlinear optimization problems nor does it rely on the construction of a low-order approximation. The method has minimal complexity and is numerically demonstrated to maintain high-order accuracy. The performance of the method is illustrated on the radiation transport equation. 

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: (1) pressure Schur complement; (2) fully coupled system using an exact factorization as a basis for the preconditioner; (3) fully coupled system using norm equivalence considerations as a basis for the preconditioner; (4) in all the cases we also investigate the benefits of the augmented Lagrangian formulation. Our objective is to see whether one of these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (the throughput is the ratio of the number of degrees of freedom of the problem divided by the number of cores and the wall-clock time in 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) The throughputs of all the methods tested in the paper are similar up to mesh-independent multiplicative constants (see Fig. 6). (2) Although very good parallel scalability is observed for the augmented Lagrangian version of the generalized Stokes problem, the best throughputs are achieved without the augmented Lagrangian term. (3) The throughput of all the methods tested in the paper is on average 5 to 25 times slower than that of traditional pressure-correction and velocity-correction methods (on average 5 for the best one). Hence, although all these methods are very efficient for solving steady state problems, pressure- 

We design pairs of six-stage, third-order, alternating implicit Runge–Kutta (RK) schemes that can be used to integrate in time two stiff operators by an operator-splitting technique. We also design for each pair a companion explicit RK scheme to be used for a third, nonstiff oper- ator in an implicit-explicit (IMEX) fashion. The main application we have in mind is (non)linear parabolic problems, where the two stiff operators represent diffusion processes (for instance, in two spatial directions) and the nonstiff operator represents (non)linear transport. We identify necessary conditions for linear sectorial A( )-stability by considering a scalar ODE with two (complex) ei- genvalues lying in some fixed cone of the half-complex plane with nonpositive real part. We show numerically that it is possible to achieve A(0)-stability when combining two operators with negative eigenvalues, irrespective of their relative magnitude. Finally, we show by numerical examples includ- ing two-dimensional nonlinear transport problems discretized in space using finite elements that the proposed schemes behave well. 

In this paper we construct an explicit approximation for the Lagrangian hydrodynamics equations equipped with an arbitrary equation of state. The approximation of the state variable is done with piecewise constant finite elements and the approximation of the mesh motion is done with higher-order continuous finite elements. The method is invariant-domain preserving and locally mass conservative. The purpose of this method is to be used in combination with higher-order methods to make them invariant domain preserving as well. 

The paper analyzes the discontinuous Galerkin approximation of Maxwell’s equations written in first-order form and with nonhomogeneous magnetic permeability and electric permittivity. Although the Sobolev smoothness index of the solution may be smaller than 1 2 , it is shown that the approximation converges strongly and is therefore spectrally correct. The convergence proof uses the notion of involution and is based on a deflated inf-sup condition and a duality argument. One essential idea is that the smoothness index of the dual solution is always larger than 1 2 irrespective of the regularity of the material properties. 

We propose an operator-splitting scheme to approximate scalar conservation equations with stiff source terms having multiple (at least two) stable equilibrium points. The scheme com- bines a (reaction-free) transport substep followed by a (transport-free) reaction substep. The transport substep is approximated using the forward Euler method with continuous finite elements and graph viscosity. The reaction substep is approximated using an exponential integrator. The crucial idea of the paper is to use a mesh-dependent cutoff of the reaction time-scale in the reaction substep. We establish a bound on the entropy residual motivating the design of the scheme. We show that the proposed scheme is invariant-domain preserv- ing under the same CFL restriction on the time step as in the nonreactive case. Numerical experiments in one and two space dimensions using linear, convex, and nonconvex fluxes with smooth and nonsmooth initial data in various regimes show that the proposed scheme is asymptotic preserving. 

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. 

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. 

The discontinuous Galerkin approximation of the grad-div and curl-curl problems formulated in conservative first-order form is investigated. It is shown that the approximation is spectrally correct, thereby confirming numerical observations made by various authors in the literature. This result hinges on the existence of discrete involutions which are formulated as discrete orthogonality properties. The involutions are crucial to establish discrete versions of weak Poincar´e–Steklov inequalities that hold true at the continuous level.