Design and Verification of the MPFA Scheme for Three-Dimensional Phase Field Model of Dendritic Crystal Growth

As an alternative to the sharp interface formulation, the phase field approach is a widely used technique for modeling phase transitions. The governing system of reaction-diffusion equations captures the instability of the underlying physical problem and is capable of modeling the evolution of complicated crystal shapes during solidification of an undercooled melt. For its numerical solution, we propose our novel anti-diffusive multipoint flux approximation (MPFA) finite volume scheme on a Cartesian mesh. The scheme is verified against the analytical solution of the modified sharp interface model. Experimental order of convergence (EOC) is measured for the temperature field in the usual norms. In addition, EOC is also obtained for the phase interface through approximating the volume of the symmetric difference of the solid phase subdomains. In the anisotropic cases including unusual higher order symmetries, computational studies with various settings also confirm convergence of our MPFA scheme which is faster than in the case of the reference finite volume scheme with 2nd order flux approximation.