Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems

Purpose - The purpose of this paper is to develop a new local radial basis function collocation method (LRBFCM) for one-domain solving of the non-linear convection-diffusion equation, as it appears in mixture continuum formulation of the energy transport in solid-liquid phase change systems. Design/methodology/approach - The method is structured on multiquadrics radial basis functions. The collocation is made locally over a set of overlapping domains of influence and the time stepping is performed in an explicit way. Only small systems of linear equations with the dimension of the number of nodes in the domain of influence have to be solved for each node. The method does not require polygonisation (meshing). The solution is found only on a set of nodes. Findings - The computational effort grows roughly linearly with the number of the nodes. Results are compared with the existing steady analytical solutions for one-dimensional convective-diffusive problem with and without phase change. Regular and randomly displaced node arrangements have been employed. The solution is compared with the results of the classical finite volume method. Excellent agreement with analytical solution and reference numerical method has been found. Practical implications - A realistic two-dimensional non-linear industrial test associated with direct-chill, continuously cast aluminium alloy slab is presented. Originality/value - A new meshless method is presented which is simple, efficient, accurate, and applicable in industrial convective-diffusive solid-liquid phase-change problems with non-linear material properties.