Quantized State Simulation of Advection-Diffusion-Reaction Equations

Abstract

Time-dependent Advection-Diffusion-Reaction (ADR) equations are used in areas such as chemistry, physics and engineering. These areas include chemical reactions, population dynamics, flame propagation, and the evolution of concentrations in environmental and biological processes.
Each of the three phenomena (advection, diffusion, and reaction) evolves in a different time scale, thus the model shows a stiff behavior.
This equation is usually discretized along the spatial variables using a grid, converting it into a large sparse set of ordinary differential equations (ODEs) that can be then solved using numerical integration methods that discretize the time variable.
An alternative way is the usage of Quantized State Systems (QSS) methods, a family of numerical integration algorithms that replace the time discretization by the quantization of the state variables. Some QSS algorithms can efficiently integrate sparse stiff ODEs, which makes them promising candidates for the ADR problem.
In this article we study the use of QSS methods for ADR models semi–discretized with the Method Of Lines. We compare the performance and the quality of the solutions obtained by these algorithms with those of conventional methods, such as DASSL, Radau and DOPRI.
Analyzing simulation times we show that, in most situations, the second order linearly implicit QSS method (LIQSS2) outperforms all the conventional algorithms in more than one order of magnitude.