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An explicit-implicit Generalized Finite Difference scheme for a parabolic-elliptic density-suppressed motility system

Federico Herrero-Hervás

TL;DR

This work develops an explicit-implicit Generalized Finite Difference (GFD) scheme for a parabolic-elliptic density-suppressed motility system, enabling meshless computation on irregular domains. The method builds spatial derivatives via a weighted least-squares fit on $E_s$-stars, yielding a convergent scheme under a time-step bound $\Delta t$, and couples a parabolic equation for $u$ with an elliptic equation for $v$ through a two-step update. A rigorous convergence theorem is provided, and an implementable algorithm is described. Numerical tests on regular and irregular grids demonstrate robustness and accuracy, with solutions converging to the carrying-capacity state and showing close agreement between grid types.

Abstract

In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic-elliptic system modelling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit-implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.

An explicit-implicit Generalized Finite Difference scheme for a parabolic-elliptic density-suppressed motility system

TL;DR

This work develops an explicit-implicit Generalized Finite Difference (GFD) scheme for a parabolic-elliptic density-suppressed motility system, enabling meshless computation on irregular domains. The method builds spatial derivatives via a weighted least-squares fit on -stars, yielding a convergent scheme under a time-step bound , and couples a parabolic equation for with an elliptic equation for through a two-step update. A rigorous convergence theorem is provided, and an implementable algorithm is described. Numerical tests on regular and irregular grids demonstrate robustness and accuracy, with solutions converging to the carrying-capacity state and showing close agreement between grid types.

Abstract

In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic-elliptic system modelling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit-implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.
Paper Structure (9 sections, 1 theorem, 34 equations, 9 figures, 2 tables, 3 algorithms)

This paper contains 9 sections, 1 theorem, 34 equations, 9 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Explicit-implicit GFD Scheme
  3. Brief description of the GFD Method
  4. Scheme and algorithm for the density-suppressed motility model
  5. Convergence of the method
  6. Numerical Tests
  7. Example 1
  8. Example 2
  9. Conclusions

Key Result

Theorem 2.1

Let $(u,v)$ be the solution to system 1.2 under conditions 1.3-1.4, then the GFD scheme 2.9-2.10 is convergent if the time step $\Delta t$ is such that holds for every inner node, where coefficients $A'_1$, $A"_1$ and $B_1$ are stated in the proof.

Figures (9)

  • Figure 1: Representation of the nodes that make up an $E_6-$star centered in $\boldsymbol{z_j}$.
  • Figure 2: A boundary node (colored in blue) and an inner node (in red) placed along the direction of the normal vector.
  • Figure 3: Discretizations of $\Omega$ considered: regular mesh of 19 inner nodes (left) and irregular cloud of points (right).
  • Figure 4: Initial condition for Example 1.
  • Figure 5: Numerical solution $(U,V)$ at $t = 0.05$ using the regular mesh (upper images) and the irregular cloud of points (lower images).
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 2.1