Numerical analysis of a time filtered scheme for a linear hyperbolic equation inspired by DNA transcription modeling

Abstract

The focus of this paper is the development and analysis of a time filtering process for a linear hyperbolic equation motivated by the modeling of the transcription of ribosomal RNA in bacteria Davis et al. (2021). We demonstrate that a time filter technique can be combined with the classical upwind to produce a new explicit scheme with virtually no dissipation introduced by the method, and the filter can be implemented with minimal computational cost. The analysis shows that the filtered scheme gives the practitioner the ability to adjust the filtering so the dissipation can be made arbitrarily small over a range of time step choices. The analysis also indicates that the filtered scheme has a smaller local truncation error when compared to that of the original upwind method. A CFL condition for the new algorithm is derived, and it is shown to depend explicitly on the filter parameter. Numerical computations illustrate stability and convergence as well as dissipation and dispersion assessments of the filtered upwind scheme.