Implicit-explicit-compact methods for advection diffusion reaction equations

TL;DR

This paper compares various spatiotemporal discretization schemes for 1D advection-diffusion-reaction equations, highlighting the superior dispersion and stability properties of the IMEX-NCCD scheme through theoretical analysis and numerical experiments.

Contribution

It introduces and evaluates the IMEX-NCCD scheme, demonstrating its enhanced stability and dispersion preservation over existing methods for ADR equations.

Findings

IMEX-NCCD has a larger neutral stability region.

IMEX-NCCD exhibits minimal dispersion error.

IMEX-NCCD reduces unphysical oscillations on coarse grids.

Abstract

We provide a preliminary comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity and the error in the phase speed of four spatiotemporal discretization schemes utilized for solving the one-dimensional (1D) linear advection diffusion reaction (ADR) equation: (a) An explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3) and the central difference scheme (CD2) for second order spatial discretization, (b) a fully implicit mid-point rule for time integration coupled with the OUCS3 and the Lele's compact scheme for first and second order spatial discretization, respectively, (c) An implicit (mid-point rule)-explicit (RK2) or IMEX time integration blended with OUCS3 and Lele's compact scheme (where the IMEX time integration follows the same ideology as introduced by Ascher et al.), and (d) the IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme). Analysis reveal the superior resolution features of the IMEX-NCCD scheme including an enhanced region of neutral stability (a region where the amplification factor is close to one), a diminished region of spurious propagation characteristics (or a region of negative group velocity) and a smaller region of nonzero phase speed error. The dispersion error of these numerical schemes through the role of q-waves is further investigated using the novel error propagation equation for the 1D linear ADR equation. Again, the in silico experiments divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs' phenomena) on coarser grids.