Numerical schemes for the solution of non-linear moisture transfer in porous materials: implicit or explicit? That is the question!
Résumé
Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating non-linear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort-Frankel, Crank-Nicolson and hyperbolisation approaches. A first case study has been considered with the hypothesis of linear transfer. The Dufort-Frankel, Crank-Nicolson and hyperbolisation schemes were compared to the classical Euler explicit scheme and to a reference solution. Results have shown that the hyperbolisation scheme has a stability condition higher than the standard Courant-Friedrichs-Lewy (CFL) condition. The error of this schemes depends on the parameter \tau representing the hyperbolicity magnitude added into the equation. The Dufort-Frankel scheme has the advantages of being unconditionally stable and is preferable for non-linear transfer, which is the second case study. Results have shown the error is proportional to O(\Delta t). A modified Crank-Nicolson scheme has been proposed in order to avoid sub-iterations to treat the non-linearities at each time step. The main advantages of the Dufort-Frankel scheme are (i) to be twice faster than the Crank-Nicolson approach; (ii) to compute explicitly the solution at each time step; (iii) to be unconditionally stable and (iv) easier to parallelise on high-performance computer systems. Although the approach is unconditionally stable, the choice of the time discretisation $\Delta t$ remains an important issue to accurately represent the physical phenomena.
Domaines
Mécanique des fluides [physics.class-ph] Physique Numérique [physics.comp-ph] Analyse numérique [cs.NA] Modélisation et simulation Logiciel mathématique [cs.MS] Equations aux dérivées partielles [math.AP] Analyse numérique [math.NA] Matériaux Thermique [physics.class-ph] Calcul parallèle, distribué et partagé [cs.DC] Algorithme et structure de données [cs.DS]
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