Realizable High-Order Methods for Non-Equilibrium, Multi-Physics Flows
|Most real-life flows are multi-physics, meaning that they encompass several different physical phenomena, in addition to the mechanism of the basic flow. Numerically solving multi-physics problems is not trivial. The major desired properties of any numerical scheme are accuracy, efficiency, and realizability.
In this study, we focus on accuracy and efficiency, and introduce, analyze, and apply a novel semi-discrete numerical approach for solving conservation equations, applicable to both reactive and non-reactive fluid flows across a variety of one-dimensional problems. Commonly, literature refers to semi-discrete methods as discretization schemes that discretize spatial derivatives while leaving the temporal derivative continuous. These methodologies often include the method of lines or exponential integrator methods. The resulting system of ordinary differential equations (ODEs) is then solved using a time marching method.
Our innovative approach discretizes the temporal derivative, leaving the spatial derivatives continuous. Subsequently, the ODE is solved analytically in the local region surrounding each finite difference mesh point, resulting in a set of algebraic tridiagonal equations that can be straightforwardly solved numerically. The primary advantage of this method is the attainment of high spatial accuracy with minimal computational cost.
We will present this new method in comprehensive detail, utilizing various discretization methods for the temporal derivative, and analyzing it in terms of stability and the order of convergence. We will then apply it to a variety of problems, including systems of coupled Burgers’ equations and reactive advection-diffusion equations.