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A structure-preserving implicit exponential time differencing scheme for Maxwell-Amp`ere Nernst-Planck model

Yunzhuo Guo, Qian Yin, Zhengru Zhang

Abstract

The transport of charged particles, which can be described by the Maxwell-Ampere Nernst-Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical scheme for the MANP model in this work. The Nernst-Planck equations are treated by the implicit exponential time differencing method associated with the Slotboom transform to preserve the positivity of the concentrations. In order to be effective with the Fast Fourier Transform, additional diffusive terms are introduced into Nernst-Planck equations. Meanwhile, the correction is introduced in the Maxwell-Ampere equation to fulfill Gauss's law. The curl-free condition for electric displacement is realized by a local curl-free relaxation algorithm whose complexity is O(N). We present sufficient restrictions on the time and spatial steps to satisfy the positivity and energy dissipation law at a discrete level. Numerical experiments are conducted to validate the expected numerical accuracy and demonstrate the structure-preserving properties of the proposed method.

A structure-preserving implicit exponential time differencing scheme for Maxwell-Amp`ere Nernst-Planck model

Abstract

The transport of charged particles, which can be described by the Maxwell-Ampere Nernst-Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical scheme for the MANP model in this work. The Nernst-Planck equations are treated by the implicit exponential time differencing method associated with the Slotboom transform to preserve the positivity of the concentrations. In order to be effective with the Fast Fourier Transform, additional diffusive terms are introduced into Nernst-Planck equations. Meanwhile, the correction is introduced in the Maxwell-Ampere equation to fulfill Gauss's law. The curl-free condition for electric displacement is realized by a local curl-free relaxation algorithm whose complexity is O(N). We present sufficient restrictions on the time and spatial steps to satisfy the positivity and energy dissipation law at a discrete level. Numerical experiments are conducted to validate the expected numerical accuracy and demonstrate the structure-preserving properties of the proposed method.
Paper Structure (16 sections, 6 theorems, 118 equations, 2 tables, 1 algorithm)

This paper contains 16 sections, 6 theorems, 118 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spatial Discretization
  3. Basic definitions
  4. Discrete operators, inner products, and norms
  5. Numerical method
  6. The implicit ETD method for Nernst-Planck equations
  7. The numerical scheme for Maxwell-Ampère equation
  8. Local curl-free algorithm
  9. Analysis on structure-preserving properties
  10. Unique solvability
  11. Mass conservation
  12. Positivity-preserving property
  13. Energy dissipation
  14. Numerical test
  15. Convergence rate
  16. ...and 1 more sections

Key Result

lemma thmcounterlemma

FunctionsMatrices Assume $\phi$ is defined on the spectrum of matrix $A \in \mathbb{C}^{m \times m}$, that is, the values exist, where $\{ \lambda_i \}_{i=1}^m$ are the eigenvalues of $A$ and $n_i$ is the order of the largest Jordan block where $\lambda_i$ appears. Then

Theorems & Definitions (18)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • ...and 8 more