High Order Stable Methods For Integer And Fractional Order Phase-field Equations

This thesis is dedicated to study and analysis of high-order stable methods for in-teger and fractional order phase-field equations.Phase-field approach has been demon-strated successful in simulating interface problems.Efficient and energy stable numeri-cal schemes for phase-field equations are very prevalent in the last few decades.In this thesis,we will combine the scalar auxiliary variable method and discontinuous Galerkin(DG)method to obtain fully discrete schemes for the phase-field equations,which can be of arbitrarily high order and satisfying energy stability.Due to the weak singularity,non-locality and multi-scale behaviors of the time-fractional phase-field(TFPF)equations,it is very difficult to directly study high-order stable schemes for the TFPF equations.In this thesis,we first investigate the proper-ties for the L2-1_σ,L2 and fast L2-1_σdiscrete fractional-derivative operators on genneral nonuniform meshes.Then the positive definiteness for the bilinear forms associated with the three operators is examined,which is the core issue in constructing the energy stable schemes for TFPF equations.Based on the positive definiteness results,the long time H~1-stabilities of high order methods are then derived for subdiffusion equations.Finally,some high-order stable methods for TFPF equations are provided.This thesis is organized as follows.Chapter 1 presents the research background and current development,as well as the main contents of this thesis.We introduce the background of integer and fractional order phase field equations.The difficulties in studying high order stable methods for the phase-field equations are explained.Chapter 2 constructs and analyzes a stable higher order fully discrete method for phase-field equations,which uses extrapolated Runge–Kutta with scalar auxiliary vari-able(RK-SAV)method in time and DG method in space.A new technique is proposed to decouple the system,after which only several elliptic scalar problems with constant coefficients need to be solved independently.Discrete energy diminishing property of the method is proved.The scheme can be of arbitrarily high order both in time and space,which is demonstrated rigorously for the Allen–Cahn equation and the Cahn–Hilliard e-quation.More precisely,optimal L~2-error bound in space and qth-order convergence rate in time are obtained for q-stage extrapolated RK-SAV/DG method.Chapter 3 focus on high order methods for time-fractional problems.The mono-tonicity of the L2-1_σ,L2 and fast L2-1_σcoefficients for Caputo derivative is provided and proved first.Then under some mild restrictions on time stepsize,the positive definiteness of the bilinear forms associated with the L2-1_σ,L2 and fast L2-1_σformulas is proved.As a consequence,the uniform global-in-time H~1-stability of the L2-1_σ,L2 and fast L2-1_σschemes can be derived for subdiffusion equations,in the sense that the H~1-norm is uniformly bounded as the time tends to infinity.In addition,the sharp error analysis is proved for the L2-1_σand fast L2-1_σmethods on general nonuniform meshes for subdiffu-sion equations,where the restriction on time step ratios is relaxed comparing to existing works.The sub-optimal H~1-norm error analysis is provied for L2 method of subdiffusion equations on general nonuniform meshes.To the best of our knowledge,this is the first work on the H~1-norm convergence for L2 method on general nonuniform meshes.In the end of this Chapter,the roundoff error problems are investigated,which have troubled re-searchers for a long time.The roundoff issue occurred frequently in interpolation methods of time-fractional equations,which can lead to wrong results in simulations.These prob-lems are essentially caused by catastrophic cancellations.In this Chapter,a new frame-work to handle catastrophic cancellations is proposed,in particular,in the computation of the coefficients for standard and fast L2-type methods on general nonuniform mesh-es.A concept ofδ-cancellation is proposed with some threshold conditions which ensure thatδ-cancellations will not happen.If the threshold conditions are not satisfied,then a Taylor-expansion technique is proposed to avoidδ-cancellation.In Chapter 4,based on the properties of fractional discrete operators studied in Chap-ter 3,high-order stable numerical methods for TFPF equations are provieded.The fast L2-1_σscheme based on second order linearization is considered first.Then a fast uncon-ditional energy stable method is constructed for TFPF equations by combining the SAV method and the fast L2-1_σapproximation.