On the geometric evolution and second order dynamics of level-sets of fields defined by partial differential equations

Materials and Mechanical Engineering, University of Oulu, Oulu, Finland

^* Corresponding author: Aarne.Pohjonen@Oulu.fi; Aarne.Pohjonen@iki.fi

Received: 4 September 2024
Accepted: 20 August 2025

Abstract

Mathematical analysis of the physical principles governing phase morphology evolution is important for the fundamental understanding of the microstructure formation in materials science. Since the physical laws describing the full field evolution of phases can be written as partial differential equations (PDEs), it is of interest to examine the general principles of fields evolution as defined by PDEs. Previously, the connection between first order time partial differential equations and the movement of level-sets (i.e. iso-contours or iso-surfaces) of a general field was established, and demonstrations of the initial calculation with a lagrangian mesh based on moving level-sets was made [A. Pohjonen, J. Phys. Conf. Ser. 2675 (2023) 012031]. Although the first order approximation is capable of estimating field evolution, the stability of numerical methods could be improved by considering second order approximations in time. Furthermore, the theoretical connection between second order time derivative of the field, the differential geometry of the field, velocity and acceleration of level-sets, and their shape evolution provide insights in to the dynamical evolution of the field geometry, as defined by partial differential equations in general.