Time discretization versus state quantization in the simulation of a one-dimensional advection–diffusion–reaction equation

Abstract

In this article, we study the effects of replacing the time discretization by the quantization of the state variables on a one dimensional Advection-Diffusion-Reaction (ADR) problem. For that purpose the 1D ADR equation is first discretized in space using a regular grid, to obtain a set of time dependent ordinary differential equations (ODEs). Then we compare the simulation performance using classic discrete time algorithms and using Quantized State Systems (QSS) methods. The performance analysis is done for different sets of diffusion and reaction parameters and also changing the space discretization refinement. This analysis shows that, in advection-reaction dominated situations, the second order linearly implicit QSS method outperforms all the conventional algorithms (DOPRI, Radau and DASSL) in more than one order of magnitude.

Improving Linearly Implicit Quantized State System Methods

“…As was already reported in Bergero et al, 15 LIQSS2 overperforms DOPRI and DASSL. However, the modified version of LIQSS2 presented in this work is more than two times faster than the original one, extending the previously reported advantages.…”

“…ADR problems discretized with the method of lines lead to large stiff systems of ODEs where the use of LIQSS methods have shown important advantages over classic discrete time algorithms. 15…”

“…The following set of ODEs, taken from Bergero et al, 15 corresponds to the spatial discretization of a 1D ADR problem:…”

“…x j //store previous value of dxj/dt 63 _ x j = f j (q(t),t) // recompute state derivative 64 € x j = _ f j (q(t),t) // recompute state second derivative 65 t η j = min(τ > t) subject to x j (τ) = q j or q j À x j (τ) = 2 Q j // recompute the time of the next change in the j-th quantized state 66 15 The following set of ODEs, taken from Bergero et al, 15 corresponds to the spatial discretization of a 1D ADR problem:…”

Co-simulation of cyber-physical systems using a DEVS wrapping strategy in the MECSYCO middleware

“…This approach is based on state quantization instead of the time discretization used by traditional integration methods. This strategy shows in some cases [24] better performances than traditional methods [25]. QSS is well-suited for hybrid modeling as it makes the continuous component equivalent to a DEVS model, which naturally integrates input events, and makes state-events detection trivial and costless [26].…”