Numerical Analysis of a Bio-Polymerization Model

TL;DR

This paper introduces a stabilization technique for hyperbolic differential equations in DNA transcription modeling, addressing oscillations in finite element methods through filtering, with proven stability and demonstrated biological relevance.

Contribution

It proposes a novel stabilization method involving filtering for hyperbolic PDEs in DNA transcription models, with theoretical convergence proofs and computational validation.

Findings

Stability and convergence of the proposed method are established.

Numerical experiments demonstrate accurate rates in space and time.

The method effectively handles biological scenarios with nonsmooth solutions.

Abstract

This work studies a stabilization technique for first-order hyperbolic differential equations used in DNA transcription modeling. Specifically we use the Lighthill-Whitham-Richards Model with a nonlinear Greenshield's velocity proposed in [1]. Standard finite element methods are known to produce spurious oscillations when applied to nonsmooth solutions. To address this, we incorporate stabilization terms involving spatial and temporal filtering into the system. We present numerical stability and prove convergence results for both the backwards Euler and time filtered formulations. We also present several computational results to demonstrate the rates in space and in time as well as for selected biological scenarios.