Collaborative Research: Efficient, Accurate, and Structure-Preserving Numerical Methods for Phase Fields-Type Models with Applications

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The project will develop computational methods for simulations of phase transformations in materials at the atomic and nanometer scales, aiming at understanding behavior at large time scales. With these simulations, the project will contribute to the understanding of processes such as complex biological growth and cancer, multi-phase active-particle and ionic fluids relevant in biological growth and development, and in the study of other complex phenomena in physics, and material engineering. The focus will be on a particular class of models: gradient flow equations with singular energy potentials. The project will develop theory and software; the codes developed in this project will be scaled up to conduct real-world three-dimensional simulations. In addition, some numerical algorithms to be developed could impact the field of deep learning. This project will provide interdisciplinary applied mathematics and scientific computing training and research experiences for both graduate and undergraduate students at the two institutions involved. In the proposed gradient flow models, a singularity is involved in the energy potential, so that the positivity-preserving property becomes a crucial feature to make the numerical approximation well-defined. In addition, energy stability and optimal rate convergence analysis will be considered for these gradient model with singular energy potential, such as the doubly degenerate Cahn-Hilliard model describing surface diffusion, a new phase field crystal model with heat transport for simulating solidification, a new quasi-incompressible Cahn-Hilliard-Navier-Stokes model for two-phase density mismatched flow, the Poisson-Nernst-Plank model for ionic mixtures, and multi-phase magneto-hydrodynamics equations. Novel finite difference, mixed finite element, and/or Fourier pseudo-spectral spatial approximations will be utilized. Convergence analysis up to the third order temporal accuracy will be investigated in details, which will be the first such work for gradient flows with singular potential. Moreover, numerical solvers for these highly nonlinear schemes will be designed and analyzed, based on the preconditioned steepest decent and Nesterov accelerated methods. Highly efficient adaptive nonlinear multigrid methods based will also be tested and studied in details.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.