Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementing truncated Fourier–Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. A striking property of some of these schemes is that they can be proved to be both 2D well-balanced and asymptotic-preserving in the parabolic limit, even when setting up IMEX time-integrators: see Corollaries 3.4 and A.1. These findings are further confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.

The Fokker-Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive limit. In order to set up a well-balanced discretization relying on an S-matrix, exact steady states of the continuous equation are derived. The ability of the scheme to keep these stationary solutions invariant produces the discretization of the local differential operator which mimics the collision process. The aforementioned scheme can be reformulated as an implicit-explicit one, which is proved to be both well-balanced and asymptotic-preserving in the diffusion limit. Several numerical benchmarks, computed on coarse grids, are displayed so as to illustrate the results.

Dissipative kinetic models inspired by neutron transport are studied in a (1 + 1)-dimensional context: first, in the two-stream approximation, then in the general case of continuous velocities. Both are known to relax, in the diffusive scaling, toward a damped heat equation. Accordingly, it is shown that “uniformly accurate” ${\mathscr{L}}$ -splines discretizations of this parabolic asymptotic equation emerge from well-balanced schemes involving scattering S-matrices for the kinetic models. Moreover, well-balanced properties are shown to be preserved when applying IMEX time-integrators in the diffusive scaling. Numerical tests confirm these theoretical findings.

A correction to this paper has been published: https://doi.org/10.1007/s42985-021-00091-x

This article proposes an efficient explicit numerical model with a relaxed stability condition for the simulation of heat, air, and moisture transfer in porous material. Three innovative approaches are combined to solve the system of two differential advection‐diffusion equations coupled with a purely diffusive equation. First, the DU FORT‐FRANKEL scheme is used to solve the diffusion equation, providing an explicit scheme with an extended stability region. Then, the two advection‐diffusion equations are solved using both the SCHARFETTER‐GUMMEL numerical scheme for the space discretisation and the two‐step RUNGE‐KUTTA method for the time variable. This combination enables to relax the stability condition by one order. The proposed numerical model is evaluated on three case studies. The first one considers quasi‐linear coefficients. The theoretical results of the numerical schemes are confirmed by our computations. Indeed, the stability condition is relaxed by a factor of 40 compared to the standard EULER explicit approach. The second case provides an analytical solution for a weakly nonlinear problem. A very satisfactory accuracy is observed between the reference solution and the one provided by the numerical model. The last case study assumes a more realistic application with nonlinear coefficients and ROBIN‐type boundary conditions. The computational time is reduced 10 times by using the proposed model in comparison with the explicit EULER method.

For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly 2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [L. Gosse and G. Toscani, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 337-342] in the 1D case. Building on former results of Birkhoff and Abu-Shumays [J. Math. Anal. Appl., 28 (1969), pp. 211-221], it is possible to express 2D kinetic steady-states by means of harmonic polynomials, and this allows one to build a scattering S-matrix yielding a time-marching scheme. Such an S-matrix can be decomposed, as in [L. Gosse and N. Vauchelet, Numer. Math., 141 (2019), pp. 627-680], so as to deduce another scheme, well-suited for a diffusive approximation of the kinetic model, for which rigorous convergence can be proved. Challenging benchmarks are also displayed on coarse grids.

A notion of "2D well-balanced" for drift-diffusion is proposed. Exactness at steady-state, typical in 1D, is weakened by aliasing errors when deriving "truly 2D" numerical fluxes from local Green's function. A main ingredient for proving that such a property holds is the optimality of the trapezoidal rule for periodic functions. In accordance with practical evidence, a "Bessel scheme" previously introduced in [SIAM J. Numer. Anal. 56 (2018), pp. 2845-2870] is shown to be "2D well-balanced" (along with former algorithms known as "discrete weighted means" or "tailored schemes". Some L-2 stability estimates are established, too.

This paper is concerned with diffusive approximations of some numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov–Fokker–Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a “scattering S-matrix”, itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il’in/Scharfetter–Gummel’s “exponential fitting” discretization. We prove that these well-balanced schemes relax, within a parabolic rescaling, towards such type of discretization by means of an appropriate decomposition of the S-matrix, hence are asymptotic preserving.

Localization phenomena (sometimes called “ flea on the elephant'') for the operator $L^\varepsilon=-\varepsilon^2 \Delta u + p(\mathbf{x}) u$, $p(\mathbf{x})$ being an asymmetric double well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov--Poincaré operator for $L^\varepsilon$, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layers.

HIV caused 35 million deaths around the world. Despite the huge efforts spent in R&D in the last decades to develop new therapeutics and vaccines, the virus spread remains uncontrolled. With the last clinical trials evidencing the challenges to develop an effective prophylactic HIV vaccine, smart and innovative solutions to design vaccine formulations which overcome these immunological obstacles are still highly anticipated. In this context, we used nanostructured lipid carriers (NLC) for the delivery of p24 antigen as an HIV model antigen. We have designed tuned formulations comprising NLC grafted with p24, together with cationic NLC optimized for the delivery of CpG immunostimulant. This tailored system strongly promoted specific immune responses in mice. More importantly, the potency of NLC for the induction of specific immune responses was further supported by a study on non-human primates (NHP). This work paves the way towards the development of future HIV vaccines.

We investigate numerically a model consisting in a kinetic equation for the biased motion of bacteria following a run-and-tumble process, coupled with two reaction-diffusion equations for chemical signals. This model exhibits asymptotic propagation at a constant speed. In particular, it admits travelling wave solutions. To capture this propagation, we propose a well-balanced numerical scheme based on Case's elementary solutions for the kinetic equation, and L-splines for the parabolic equations. We use this scheme to explore the Cauchy problem for various parameters. Some examples far from the diffusive regime lead to the coexistence of two waves travelling at different speeds. Numerical tests support the hypothesis that they are both locally asymptotically stable. Interestingly, the exploration of the bifurcation diagram raises counter-intuitive features.

A genuinely two-dimensional discretization of general drift-diffusion (including incompressible Navier--Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of “bubbles” which are deduced from available discrete data by exploiting the stationary Dirichlet--Green function of the convection-diffusion operator. These fluxes are reminiscent of Scharfetter and Gummel's in the sense that they contain modified Bessel functions which allow one to pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties are established in the vanishing viscosity limit (``asymptotic monotony'') along with second-order accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the scheme, including the “western currents” with a Navier--Stokes--Coriolis model of ocean circulation.

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax–Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the H^s norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former “shape functions” and “symmetric potential schemes” are highlighted.

Well-balanced schemes, nowadays well-known for 1D hyperbolic equations with source terms and systems of balance laws, are extended to strictly parabolic equations, first in 1D, then in 2D on Cartesian computational grids. The construction heavily relies on a particular type of piecewise-smooth interpolation of discrete data known as L-splines. In 1D, several types of widely-used discretizations are recovered as particular cases, like the El-Mistikawy-Werle scheme or Scharfetter-Gummel's. Moreover, a distinctive feature of our 2D scheme is that dimensional-splitting never occurs within its derivation, so that all the multidimensional interactions are kept at the discrete level. This leads to improved accuracy, as illustrated on several types of drift-diffusion equations.

Well-balanced schemes, nowadays well-known for 1D hyperbolic equations with source terms and systems of balance laws, are extended to strictly parabolic equations, first in 1D, then in 2D on Cartesian computational grids. The construction heavily relies on a particular type of piecewise-smooth interpolation of discrete data known as \(\mathcal{L}\)-splines. In 1D, several types of widely-used discretizations are recovered as particular cases, like the El-Mistikawy-Werle scheme or Scharfetter-Gummel’s. Moreover, a distinctive feature of our 2D scheme is that dimensional-splitting never occurs within its derivation, so that all the multi-dimensional interactions are kept at the discrete level. This leads to improved accuracy, as illustrated on several types of drift-diffusion equations.

A model consisting of a kinetic equation for “run-and-tumble” biased bacteria motion, coupled with two reaction-diffusion equations for chemical signals, is studied. It displays time-asymptotic propagation at constant velocity, i.e., aggregated travelling (exponential) layers. To capture them for various parameters, a well-balanced setup is based on both “Case's elementary solutions" and $\mathcal{L}$-spline reconstruction. Far from the diffusive regime, waves travelling at different velocities (bistability) are proved to coexist. Numerics suggest that they are locally asymptotically stable, so that the resulting bifurcation diagram shows counterintuitive features.

Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a N-wave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presented.

The existence of travelling waves for a coupled system of hyperbolic/parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum of velocities. Here, the proof is specific to the discrete setting, based on the decomposition of the population density in special Case’s modes. Some counter-intuitive results are discussed numerically, including the co-existence of several travelling waves for some sets of parameters, as well as the possible non-existence of travelling waves.

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a scattering S-matrix , itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il'in/Scharfetter-Gummel's exponential fitting discretization. We prove that the well-balanced schemes relax, within a parabolic rescaling, towards the Il'in exponential-fitting discretization by means of an appropriate decomposition of the S-matrix. This is the so-called asymptotic preserving (or uniformly accurate) property.

In this note we review and recast some recent results on the existence of non-standard solutions to the compressible Euler equations as to make possible a preliminary numerical investigation. In particular, we are interested in studying numerically the forward in time evolution of some Lipschitz initial data which allow for non-standard solutions ("colliding data"). Numerical results indicate appearance of oscillations after the first break-up time along with a qualitative behavior seemingly compatible with relevant properties of non-standard solutions.