Numerical Analysis of a Bio-Polymerization Model

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.