Ion transport, often described by the Poisson--Nernst--Planck (PNP) equations, is ubiquitous in electrochemical devices and many biological processes of significance. In this work, we develop conservative, positivity-preserving, energy dissipating, and implicit finite difference schemes for solving the multi-dimensional PNP equations with multiple ionic species. A central-differencing discretization based on harmonic-mean approximations is employed for the Nernst--Planck (NP) equations. The backward Euler discretization in time is employed to derive a fully implicit nonlinear system, which is efficiently solved by a newly proposed Newton's method. The improved computational efficiency of the Newton's method originates from the usage of the electrostatic potential as the iteration variable, rather than the unknowns of the nonlinear system that involves both the potential and concentration of multiple ionic species. Numerical analysis proves that the numerical schemes respect three desired analytical properties (conservation, positivity preserving, and energy dissipation) fully discretely. Based on advantages brought by the harmonic-mean approximations, we are able to establish estimate on the upper bound of condition numbers of coefficient matrices in linear systems that are solved iteratively. The solvability and stability of the linearized problem in the Newton's method are rigorously established as well. Numerical tests are performed to confirm the anticipated numerical accuracy, computational efficiency, and structure-preserving properties of the developed schemes. Adaptive time stepping is implemented for further efficiency improvement. Finally, the proposed numerical approaches are applied to characterize ion transport subject to a sinusoidal applied potential.
In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state $S_{N}$ transport equation in the finite volume framework. It is well-known that the $S_{N}$ transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonst…
Asymptotic preserving schemes on distorted meshes for Friedrichs systems with sti relaxation: application to angular models in linear transport
CEA, DAM, DIF（法国原子能和替代能源委员会、国防事务局、信息与通信技术部）；Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie（雅克-路易·利翁斯实验室，皮埃尔和玛丽·居里大学）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）
We introduce a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation. The proposed numerical method uses the low rank approximation only for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second order accurate and asymptotic-preserving (AP); that is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation and is automatically low ra…
This paper is devoted to the numerical approximation of the spatially-extended FitzHugh-Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude of the interactions becomes large, this method provides a consistent discretization of the macroscopic reaction-diffusion FitzHugh-Nagumo system. We carry out some theoretical proofs and perform s…
In this paper we develop a neural network for the numerical simulation of time-dependent linear transport equations with diffusive scaling and uncertainties. The goal of the network is to resolve the computational challenges of curse-of-dimensionality and multiple scales of the problem. We first show that a standard Physics-Informed Neural Network (PINN) fails to capture the multiscale nature of the problem, hence justifies the need to use Asymptotic-Preserving Neural Networks (APNNs). We show that not all classical AP formulations are fit for the neural network approach. We construct a micro-…
Bound-Preserving Flux-Corrected Transport Methods for Solving Richards' Equation
Department of Civil and Environmental Engineering, Louisiana State University（路易斯安那州立大学土木与环境工程系）；TU Dortmund University（多特蒙德工业大学）；Center for Computation and Technology, Louisiana State University（路易斯安那州立大学计算与技术中心）
We present compatible finite element space discretizations for the ideal compressible magnetohydrodynamic equations. The magnetic field is considered both in div- and curl-conforming spaces, leading to a strongly or weakly preserved zero-divergence condition, respectively. The equations are discretized in space such that transfers between the kinetic, internal, and magnetic energies are consistent, leading to a preserved total energy. We also discuss further adjustments to the discretization required to additionally achieve magnetic helicity preservation. Finally, we describe new transport sta…
Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures
Duke University（杜克大学）；University of Washington（华盛顿大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）；Georgia Institute of Technology（佐治亚理工学院）
In this paper, uniformly unconditionally stable first and second order finite difference schemes are developed for kinetic transport equations in the diffusive scaling. We first derive an approximate evolution equation for the macroscopic density, from the formal solution of the distribution function, which is then discretized by following characteristics for the transport part with a backward finite difference semi-Lagrangian approach, while the diffusive part is discretized implicitly. After the macroscopic density is available, the distribution function can be efficiently solved even with a…
We develop a new finite volume method using unstructured mesh-vertex grids for coupled systems modeling shallow water flows and solute transport over complex bottom topography. Novel well-balanced positivity preserving discretization techniques are proposed for the water surface elevation and the concentration of the pollutant. For the hydrodynamic system, the proposed scheme preserves the steady state of a lake at rest and the positivity of the water depth. For the scalar transport equation, the proposed method guarantees the positivity and a perfect balance of the scalar concentration. The c…
