# Stable explicit schemes for simulation of nonlinear moisture transfer in porous materials

## TL;DR

This paper investigates improved explicit numerical schemes like Dufort-Frankel, Crank-Nicolson, and hyperbolisation for simulating nonlinear moisture transfer in porous materials, aiming to enhance stability and computational efficiency over traditional implicit methods.

## Contribution

It introduces and compares explicit schemes for nonlinear moisture transfer, highlighting the advantages of Dufort-Frankel in stability, speed, and parallelization, with a modified Crank-Nicolson scheme to handle non-linearities.

## Key findings

- Dufort-Frankel scheme is unconditionally stable and faster than Crank-Nicolson.
- Hyperbolisation scheme has a higher stability condition than CFL.
- Modified Crank-Nicolson avoids sub-iterations for non-linear problems.

## Abstract

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.