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STRUCTURE PRESERVING NUMERICAL METHODS FOR POISSON-NERNST-PLANCK-NAVIER-STOKES SYSTEM AND GRADIENT FLOW OF OSEEN-FRANK ENERGY OF NEMATIC LIQUID CRYSTALS
This thesis consists of the structure-preserving numerical methods for PNP-NS equation and dynamic liquid crystal systems in Oseen-Frank energy.
In Chapter 1, we give a brief introduction of the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system, and the dynamical liquid system in Oseen-Frank energy in one-constant approximation case and a special non-one-constant case. Each of those systems has a special structure and properties we want to keep at the discrete level when designing numerical methods.
In Chapter 2, we introduce a first-order numerical scheme for the PNP-NS system that is decoupled, positivity preserving, mass conserving, and unconditionally energy stable. The numerical scheme is designed in the context of Wasserstein gradient flow based on the form ∇ · (c∇ ln c). The mobility terms are treated explicitly, and the chemical potential terms are treated implicitly so that the solution of the numerical scheme is the minimizer of a convex functional, which is the key to the unique solvability and positivity preserving of the numer-
ical scheme. Proper boundary conditions for chemical potentials are chosen to guarantee the mass-conservation property. The convection term in Poisson-Nernst-Planck(PNP) equation part is treated explicitly with an O(∆t) term introduced so that the numerical scheme is decoupled and unconditionally energy stable. Pressure correction methods are used for the Navier-Stokes(NS) equation part. And we proved the optimal convergence rate with an irregular high-order asymptotic expansion technique.
In Chapter 3, we propose a first-order implicit numerical method for a dynamic liquid crystal system in a one-constant-approximation case(which is also known as heat flow of harmonic maps to S2). The solution is the minimizer of a convex functional under the unit length constraint, and from this point, the weak convergence of the numerical scheme could be proved. The numerical scheme is solved in an iterative procedure. This procedure could be proved to be energy decreasing and this implies the convergence of the algorithm.
In Chapter 4, we study the dynamic liquid crystal system in a more generalized Oseen- Frank energy compared to Chapter 3. We are assuming K2 = K3 = −K4, the domain Ω is a rectangular region in R3, and d satisfies the periodic boundary condition on ∂Ω. And we propose a class of numerical schemes for this system that preserve the unit length constraint. The convergence of the numerical scheme has been proved under necessary assumptions. And numerical experiments are presented to validate the accuracy and demonstrate the performance of the proposed numerical scheme.