Implicit-explicit-compact methods for advection diffusion reaction equations

摘要

We provide a comparison of the dispersion properties, specifically the time-amplification factor, the scaled group velocity, the error in the phase speed and scaled numerical diffusion coefficient of four spatiotemporal discretization schemes utilized for solving a linear, one-dimensional (1D) as well as a linear/nonlinear, two-dimensional (2D) advection diffusion reaction (ADR) equation: (a) Explicit (RK2) temporal integration combined with the Optimal Upwind Compact Scheme (or OUCS3, J. Comp. Phys., 192, pg. 677-694 (2003)) and the central difference scheme (CD2), (b) Fully implicit mid-point rule for time integration coupled with the OUCS3 and the Lele's compact scheme (J. Comp. Phys., 103, pg. 16-42 (1992)), (c) 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., SIAM J. Numer. Anal., 32(3), pg. 797-823 (1995)), and (d) IMEX (mid-point/RK2) time integration melded with the New Combined Compact Difference scheme (or NCCD scheme, J. Comp. Phys., 228, pg. 6150-6168 (2009)). Analysis reveals the superior resolution features of the IMEX-OUCS3-Lele scheme and the IMEX-NCCD scheme including an enhanced region of asymptotic stability (a region with numerical amplification factor less than unity), a diminished region of spurious propagation characteristics (or a region of negative scaled group velocity) and a smaller region of nonzero phase speed error. In particular, the IMEX-NCCD scheme captures the correct propagation feature (or positive scaled group velocity) in the largest possible region in the wavenumber-Courant-Friedrichs-Lewy (CFL) number, parameter space, in comparison with the other three numerical methods. The in silico experiments investigating the role of q- waves in the numerical solution of the linear, 1D ADR equation divulge excellent Dispersion Relation Preservation (DRP) properties of the IMEX-NCCD scheme including minimal dissipation via implicit filtering and negligible unphysical oscillations (or Gibbs' phenomenon) on coarser grids. The numerical solution of the 2D viscous Burgers' Equation underline the supremacy of the IMEX method with regard to handling the 'stiff' derivative terms in contrast with a fully explicit time integration method and lower computational time versus with a fully implicit scheme. The DRP resolution of the IMEX-NCCD scheme is further benchmarked by solving the classical two-dimensional (2D), Patlak-Keller-Segel (PKS) nonlinear parabolic model. Numerical results reveal that the spiky structure of the solution is oscillation free and, when compared with the Explicit-OUCS3-CD2 method, the solution is better resolved by the IMEX-NCCD method. Published by Elsevier Ltd.