Dispersion-Preserving Implicit–Explicit Numerical Methods for Reactive Flow Models

Abstract

Chemical reactions occur everywhere in both natural and artificial systems. Some of the reactions occur during the flow of a fluid (such a process is referred to as a reactive flow). Given the hazardous nature of some reactive flows, computer simulations (rather than physical experiments) are necessary for ascertaining or enhancing our understanding of such systems. The process of simulation involves mathematical and numerical modeling of the reactive flows. Mathematical models for reactive flow problems are complicated partial differential equations that often lack exact solutions, thus, numerical solutions are employed. Numerical methods must preserve almost all the relevant properties of the problem for accuracy reasons. Dispersion relations are important properties of wave propagation problems and numerical methods that satisfy them are called dispersion preserving methods. Furthermore, stiff transport models are wave propagation problems that cannot be solved efficiently with explicit methods. However, fully implicit methods are computationally expensive. A combination of implicit and explicit methods called implicit–explicit methods is usually employed to efficiently resolve stiffness. An example of problems of interest in this regard are the advection–diffusion–reaction (ADR) models. In this discussion, spectral analysis is performed on two implicit–explicit methods to ascertain their dispersion preserving abilities in order to determine their suitability for simulating general stiff reactive flow problems. The analysis shows that both implicit–explicit methods are dispersion preserving, however, one particular method is more suitable for general wave propagation problems.