https://www.ki-net.umd.edu/pubs/files/FRG-2009-Jin-Shi.Filbet-Jin.pdf

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

https://hal.archives-ouvertes.fr/hal-00986714/document

Numerical methods for kinetic equations

In many parts of the world, groundwater resources are under increasing threat from growing demands, wasteful use, and contamination. To face the challenge, good planning and management practices are needed. A key to the management of groundwater is the ability to model the movement of fluids and contaminants. The purpose of this book is to construct conceptual and mathematical models that can provide the information required for making decisions associated with the management of groundwater resources, and the remediation of contaminated aquifers. The basic approach of this book is to accurately describe the underlying physics of groundwater flow and solute transport in heterogeneous porous media, starting at the microscopic level, and to rigorously derive their mathematical representations at the macroscopic levels.

Modeling groundwater flow and contaminant transport

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the L2-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Virtual Element Methods for general second order elliptic problems on polygonal meshes

A unified gas-kinetic scheme for continuum and rarefied flows IV

The virtual element method for discrete fracture network simulations

In this work a new class of numerical methods for the BGK model of kinetic equations is presented. In principle, schemes of any order of accuracy in both space and time can be constructed with this technique. The methods proposed are based on an explicit---implicit time discretization. In particular the convective terms are treated explicitly, while the source terms are implicit. In this fashion even problems with infinite stiffness can be integrated with relatively large time steps. The conservation properties of the schemes are investigated. Numerical results are shown for schemes of order 1, 2 and 5 in space, and up to third-order accurate in time.

/pdf/implicit-explicit-schemes-for-bgk-kinetic-equations-29i5qr8fmj.pdf

Implicit—Explicit Schemes for BGK Kinetic Equations

A hybrid mortar virtual element method for discrete fracture network simulations

We investigate a new numerical approach for the computation of the three-dimensional flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex three-dimensional systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. An independent meshing process within each fracture is a very important issue for practical large-scale simulations, making mesh generation easier. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures.

/pdf/a-pde-constrained-optimization-formulation-for-discrete-2cmmsud7pk.pdf

Sandra Pieraccini

Papers

The virtual element method for discrete fracture network simulations

Implicit—Explicit Schemes for BGK Kinetic Equations

A hybrid mortar virtual element method for discrete fracture network simulations

A PDE-Constrained Optimization Formulation for Discrete Fracture Network Flows

Order preserving SUPG stabilization for the Virtual Element formulation of advection-diffusion problems