Asymptotic Preserving Research Articles

A structure and asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck model

We propose a numerical method for the Vlasov-Poisson-Fokker-Planck model written as an hyperbolic system thanks to a spectral decomposition in the basis of Hermite functions with respect to the velocity variable and a structure preserving finite volume scheme for the space variable. On the one hand, we show that this scheme naturally preserves both stationary solutions and linearized free-energy estimate. On the other hand, we adapt previous arguments based on hypocoercivity methods to get quantitative estimates ensuring the exponential relaxation to equilibrium of the discrete solution for the linearized Vlasov-Poisson-Fokker-Planck system, uniformly with respect to both scaling and discretization parameters. Finally, we perform substantial numerical simulations for the nonlinear system to illustrate the efficiency of this approach for a large variety of collisional regimes (plasma echos for weakly collisional regimes and trend to equilibrium for collisional plasmas) and to highlight its robustness (unconditional stability, asymptotic preserving properties).

Unified gas-kinetic particle method for frequency-dependent radiation transport

This paper proposes a unified gas-kinetic particle (UGKP) method for the frequency-dependent photon transport process. The photon transport is a typical multiscale process governed by the nonlinear radiative transfer equations (RTE). The flow regime of photon transport varies from the ballistic regime to the diffusive regime with respect to optical depth and photon frequency. The UGKP method is an asymptotic preserving (AP) scheme and a regime adaptive scheme. The teleportation error is significantly reduced, and the computational efficiency is remarkably improved in the diffusive regime. Distinguished from the standard multigroup treatment, the proposed UGKP method solves the frequency space in a non-discretized way. Therefore the Rosseland diffusion system can be precisely preserved in the optically thick regime. Taking advantage of the local integral solution of RTE, the distribution of the emitted photon can be constructed from its macroscopic moments. The Monte Carlo particles in the UGKP method need only be tracked before their first collision events, and a re-sampling process is performed to close the photon distribution for each time step. The high computational cost of calculating excessive scattering events can be saved, especially in the optically thick regime. The proposed UGKP method is implicit and removes the light speed constraint on the time step. The particle tracking approach combining the implicit formulation makes the proposed UGKP method an efficient solution algorithm for frequency-dependent radiative transfer problems. We demonstrate with numerical examples the capability of the proposed multi-frequency UGKP method.

An Asymptotic Preserving and Energy Stable Scheme for the Barotropic Euler System in the Incompressible Limit

An Asymptotic Preserving and Energy Stable Scheme for the Barotropic Euler System in the Incompressible Limit

Asymptotic Preserving and Uniformly Unconditionally Stable Finite Difference Schemes for Kinetic Transport Equations

Asymptotic Preserving and Uniformly Unconditionally Stable Finite Difference Schemes for Kinetic Transport Equations

An efficient asymptotic preserving Monte Carlo method for radiative transfer equations

In this work, an efficient asymptotic preserving Monte Carlo method is developed for nonlinear thermal radiative transfer equations. We derive a new approximate macroscopic equation for the radiation energy density, from an integral solution of the radiation intensity along characteristics of the microscopic equation. We will solve a coupled macro-micro system with a hybrid numerical method. The macroscopic radiation and material energy densities are updated first by using a deterministic finite volume method. The microscopic equation, with the emission source provided from the macroscopic variables, can then be efficiently solved using a particle-based Monte Carlo method. This new approach is uniformly stable, with large time steps independent of scaling parameters and the speed of light. Besides, it can be shown that the method is asymptotic preserving in the diffusive limit for an optically thick regime. Numerical experiments are performed to demonstrate the high efficiency and good performances of our proposed method.

An Asymptotic-Preserving Hybrid Angular Discretization for the Gray Radiative Transfer Equations

The aim of this paper is to construct a new numerical scheme for the nonlinear gray radiative transfer (GRT) equations, namely, the asymptotic-preserving (AP) H N T -based unified gas kinetic scheme (UGKS). The constructed scheme is obtained by combing the UGKS for spatial discretization with the hybrid H N T method for angular discretization. Since the H N T is a hybrid angular discrete method of both P N and S N methods, the current H N T -based UGKS can not only mitigate the ray effects of the S N method largely, but also suppress the oscillations of the original P N method. Furthermore, we show that the current H N T -based UGKS also inherits the AP property of UGKS. A number of one-dimensional and two-dimensional numerical experiments are presented that validate the performance of the current scheme in both optically thin and thick regimes, as well as in mitigating the ray effects. Moreover, it can capture the initial layer solution without requiring additional treatments.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics, both in time and space, which include the relaxation schemes by Jin and Xin. These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case. These kinetic models depend on a small parameter that can be seen as a “Knudsen” number. The method is asymptotic preserving in this Knudsen number. Also, the computational costs of the method are of the same order of a fully explicit scheme. This work is the extension of Abgrall et al. (2022) [3] to multi-dimensional systems. We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

A new family of semi-implicit Finite Volume/Virtual Element methods for incompressible flows on unstructured meshes

We introduce a new family of high order accurate semi-implicit schemes for the solution of nonlinear time-dependent systems of partial differential equations (PDE) on unstructured polygonal meshes. The time discretization is based on a splitting between explicit and implicit terms that may arise either from the multi-scale nature of the governing equations, which involve both slow and fast scales, or in the context of projection methods, where the numerical solution is projected onto the physically meaningful solution manifold. We propose to use a high order finite volume (FV) scheme for the explicit terms, hence ensuring conservation property and robustness across shock waves, while the virtual element method (VEM) is employed to deal with the discretization of the implicit terms, which typically requires an elliptic problem to be solved. The numerical solution is then transferred via suitable L2 projection operators from the FV to the VEM solution space and vice-versa. High order time accuracy is then achieved using the semi-implicit IMEX Runge–Kutta schemes, and the novel schemes are proven to be asymptotic preserving (AP) and well-balanced (WB). As representative models, we choose the shallow water equations (SWE), thus handling multiple time scales characterized by a different Froude number, and the incompressible Navier–Stokes equations (INS), which are solved at the aid of a projection method to satisfy the solenoidal constraint of the velocity field. Furthermore, an implicit discretization for the viscous terms is also devised for the INS model, which is based on the VEM technique. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the celerity nor on the viscous eigenvalues. A large suite of test cases demonstrates the accuracy and the capabilities of the new family of schemes to solve relevant benchmarks in the field of incompressible fluids.

High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number ε ranging from 0 to O(1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit.

An asymptotic preserving scheme for the shallow‐water equations with Manning friction using viscous correction of the HLL scheme

AbstractThe aim of this article is to propose a numerical scheme for the shallow‐water equations with a Manning's friction source term able to preserve the diffusive regime, namely the behavior of the solutions in long time and stiff friction limit. Because of the non‐usual rescaling of the friction parameter, a strongly nonlinear derivative operator is involved in the diffusive limit and the development of a numerical scheme possessing this property is not straightforward. The scheme presented here is based on a perturbed HLL scheme with additional viscous terms.

Pattern Formation of a Pathway-Based Diffusion Model: Linear Stability Analysis and an Asymptotic Preserving Method

Pattern Formation of a Pathway-Based Diffusion Model: Linear Stability Analysis and an Asymptotic Preserving Method

Accuracy and Stability Analysis of the Semi-Lagrangian Method for Stiff Hyperbolic Relaxation Systems and Kinetic BGK Model

In this paper, we develop a family of third order asymptotic-preserving (AP) and asymptotically accurate (AA) diagonally implicit Runge–Kutta (DIRK) time discretization methods for the stiff hyperbolic relaxation systems and kinetic Bhatnagar–Gross–Krook (BGK) model in the semi-Lagrangian (SL) setting. The methods are constructed based on an accuracy analysis of the SL scheme for stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid regime when the Knudsen number approaches 0. An extra order condition for the asymptotic third order accuracy in the limiting regime is derived. Linear von Neumann stability analysis of the proposed third order DIRK methods are performed to a simplified two-velocity linear kinetic model. Extensive numerical tests are presented to demonstrate the AA, AP, and stability properties of our proposed schemes.

Unified preserving properties of kinetic schemes.

The kinetic theory provides a physical basis for developing multiscale methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number ε≪1) without enforcing kinetic scale resolution. At the current stage, the main approach to analyze such a property is the asymptotic preserving (AP) concept, which aims to show whether a kinetic scheme reduces to a solver for the hydrodynamic equationsas ε→0, such as the shock capturing scheme for the Euler equations. However, the detailed asymptotic properties of the kinetic scheme are indistinguishable when ε is small but finite under the AP framework. To distinguish different characteristics of kinetic schemes, in this paper we introduce the concept of unified preserving (UP) aiming at assessing asymptotic orders of a kinetic scheme by employing the modified equationapproach and Chapman-Enskon analysis. It is shown that the UP properties of a kinetic scheme generally depend on the spatial and temporal accuracy and closely on the interconnections among three scales (kinetic scale, numerical scale, and hydrodynamic scale) and their corresponding coupled dynamics. Specifically, the numerical resolution and specific discretization of particle transport and collision determine the flow physics of the scheme in different regimes, especially in the near continuum limit. As two examples, the UP methodology is applied to analyze the discrete unified gas-kinetic scheme and a second-order implicit-explicit Runge-Kutta scheme in their asymptotic behaviors in the continuum limit.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.

On some new results on anisotropic singular perturbations of second-order elliptic operators

In this article, we deal with some problems involving a class of singularly perturbed elliptic operators. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behaviour of the semigroup generated.

An asymptotic preserving scheme for Lévy-Fokker-Planck equation with fractional diffusion limit

In this paper, we develop a numerical method for the L\'evy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other arises from long-time/small mean-free-path scaling, which introduces stiffness to the equation. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply the Chebyshev polynomial based pseudo-spectral method. To treat the multiple scales, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically. Finally, the efficiency and accuracy of our scheme are illustrated by a suite of numerical examples.

Gradient discretization of a 3D-2D-1D mixed-dimensional diffusive model with resolved interface, application to the drying of a fractured porous medium

Abstract We consider a 3D-2D-1D mixed-dimensional diffusive model in a fractured porous medium coupling the 1D model along the centerline skeleton of a tubular network, the 2D model on a network of planar fractures and the 3D model in the surrounding matrix domain. The transmission conditions are based on a potential continuity assumption at matrix fracture interfaces, and on Robin type conditions at the resolved interfaces between the tubular network and the matrix and fracture network domains. The discretization of this mixed-dimensional model is formulated in the gradient discretization framework (Droniou, J., Eymard, R. & Herbin, R. (2016) Gradient schemes: generic tools for the numerical analysis of diffusion equations. ESAIM Math. Model. Numer. Anal., 50, 749–781), which covers a large class of conforming and nonconforming schemes and provides stability and error estimates based on general coercivity, consistency and limit-conformity properties. As an example of discretization fitting this framework, the mixed-dimensional version of the vertex approximate gradient (VAG) scheme is developed. It is designed to allow nonconforming meshes at the interface between the 1D and 3D-2D domains, to provide a conservative formulation with local flux expressions and to be asymptotic preserving in the limit of high transfer coefficients. Numerical experiments are provided on analytical solutions for simplified geometries, which confirm the theoretical results. Using its equivalent finite volume formulation, the VAG discretization is extended to a drying mixed-dimensional model coupling the Richards equation in a fractured porous medium and the convection diffusion of the vapor molar fraction along the 1D domain. It is applied to simulate the drying process between an operating tunnel and a radioactive waste storage rock with explicit representation of the fractures in the excavated damaged zone.

A gas kinetic Lax–Wendroff scheme for low-speed isothermal rarefied gas flows

Previously, a gas kinetic Bhatnagar–Gross–Krook (BGK) scheme was proposed by us for incompressible flows in the continuum limits. [W. Li and W. Li, “A gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible flows,” Comput. Fluids 162, 126–138 (2018).] In the present work, we extend the gas kinetic BGK scheme to simulate low-speed isothermal rarefied nonequilibrium gas flows. This scheme is a gas kinetic Lax–Wendroff scheme (GKLWS) for the discrete velocity Boltzmann equation in the finite volume discretization framework with second-order accuracy in both time and space. As collision and transport of the molecular particles are coupled in the present GKLWS, the time step of the present method is not limited by the relaxation time, for which the present scheme is efficient for multiscale gas flows. Moreover, the present GKLWS holds the asymptotic preserving (AP) property, which ensures that both the Navier–Stokes solutions in the continuum limits and free-molecular flow solutions in the rarefied limits can be reliably obtained. To validate the accuracy and AP property of the GKLWS, several numerical benchmarks of isothermal low-speed rarefied gas flows are simulated by the present scheme. The numerical results show that the present scheme can be a reliable multiscale method for all Knudsen number low-speed isothermal gas flows.

An all Froude high order IMEX scheme for the shallow water equations on unstructured Voronoi meshes

We propose a novel numerical method for the solution of the shallow water equations in different regimes of the Froude number making use of general polygonal meshes. The fluxes of the governing equations are split such that advection and acoustic-gravity sub-systems are derived, hence separating slow and fast phenomena. This splitting allows the nonlinear convective fluxes to be discretized explicitly in time, while retaining an implicit time marching for the acoustic-gravity terms. Consequently, the novel schemes are particularly well suited in the low Froude limit of the model, since no numerical viscosity is added in the implicit solver. Besides, stability follows from a milder CFL condition which is based only on the advection speed and not on the celerity. High order time accuracy is achieved using the family of semi-implicit IMEX Runge-Kutta schemes, while high order in space is granted relying on two discretizations: (i) a cell-centered finite volume (FV) scheme for the nonlinear convective contribution on the polygonal cells; (ii) a staggered discontinuous Galerkin (DG) scheme for the solution of the linear system associated to the implicit discretization of the pressure sub-system. Therefore, three different meshes are used, namely a polygonal Voronoi mesh, a triangular subgrid and a staggered quadrilateral subgrid. The novel schemes are proved to be Asymptotic Preserving (AP), hence a consistent discretization of the limit model is retrieved for vanishing Froude numbers, which is given by the so-called “lake at rest” equations. Furthermore, the novel methods are well-balanced by construction, and this property is also demonstrated. Accuracy and robustness are then validated against a set of benchmark test cases with Froude numbers ranging in the interval Fr≈[10−6;5], hence showing that multiple time scales can be handled by the novel methods.

Numerical study of multiphase hyperbolic models

Baer–Nunziato (BN) type models introduce fundamental and widely used class of multi-fluid and multi-velocity models to describe evolution of multi-phase flows. The original BN model is supposed to describe multi-phase flows at spatial scales substantially larger than the characteristic scale of elements of the dispersed phase. Recently a number of attempts have been performed to apply it to the description of multi-phase flows with resolved, at the large spatial scale, inter-phase boundaries. A number of still not well understood issues arise in this situation, related to the complex interplay of the topics related to (i) justification of the hierarchy of BN-type models for diffuse interface type description of multi-phase flows with resolved inter-phase boundaries; (ii) the questions of “correct” definition of generalized solutions of first order quasi-linear hyperbolic systems and (iii) development of “really” path-conservative and asymptotic preserving schemes for quasilinear hyperbolic systems with stiff relaxation terms. The goal of the paper is to study numerically such interplay when BN7 model is applied to describe 1D two-phase flows. Our main target is the diffuse interface type formulations. The obtained numerical solutions are compared to analytical solutions of the “more equilibrated” BN5 equations and heterogeneous formulations for the standard Euler equation. The goal is to provide factual information for the performed tests. We also try to summarize the obtained results in a systematic way from a practical point of view. To avoid ambiguity in interpretation of the numerical results, we provide detailed description of the algorithms and perform a number of auxiliary numerical tests.