Mathematical Modelling and Numerical Simulation of Diffusive Processes in Slow Changing Domains

Abstract

Nowadays, diffusion and heat conduction processes in slow changing domains attract great attention. Slow-phase transitions and growth of biological structures can be considered as examples of such processes. The main difficulty in numerical solutions of correspondent problems is connected with the presence of two time scales. The first one is time scale describing diffusion or heat conduction. The second time scale is connected with the mentioned slow domain evolution. If there is sufficient difference in order of the listed time scale, strong computational difficulties in application of time-stepping algorithms are observed. To overcome the mentioned difficulties, it is proposed to apply a small parameter method for obtaining a new mathematical model, in which the starting parabolic initial-boundary-value problem is replaced by a sequence of elliptic boundary-value problems. Application of the boundary element method for numerical solution of the obtained sequence of problems gives an opportunity to solve the whole considered problem in slow time with high accuracy specific to the mentioned algorithm. Besides that, questions about convergence of the obtained asymptotic expansion and correspondence between initial and obtained formulations of the problem are considered separately. The proposed numerical approach is illustrated by several examples of numerical calculations for relevant problems.