Highly Efficient And Accurate Algorithms For Phase Field Models

Phase field models have been widely used in biology,material science,and engineer-ing,examples include:the growth of crystal,the epitaxial thin film growth,condensation of gas,solidification of liquid.A challenging topic is to design efficient and accurate numerical schemes for such models.In the chapters below,we will construct some fast and efficient numerical schemes for various phase field models and also provide stability analysis for such models.1.We consider numerical approximations of a hydro-dynamically coupled phase field diblock copolymer model,in which the free energy contains a kinetic potential,a gradient entropy,a Ginzburg-Landau double well potential,and a long range nonlocal type potential.We develop a set of second order time marching schemes for this system using the "Invariant Energy Quadratization"approach for the double well potential,the projection method for the Navier-Stokes equation,and a subtle implicit-explicit treatment for the stress and convective term.The resulting schemes are linear and lead to symmetric positive definite systems at each time step,thus they can be efficiently solved.We further prove that these schemes are unconditionally energy stable.Various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.2.We develop highly efficient,second order and unconditionally energy stable schenes for the epitaxial thin film growth models by using the scalar auxiliary vari-able(SAV)approach.A main difficulty here is that the nonlinear potential for the model without slope selection is not,bounded from below so the SAV approach can not be direct-ly applied.We overcome this difficulty with a suitable splitting of the total free energy density into two parts such that the integral of the part involving the nonlinear poten-tial becomes bounded from below so that the SAV approach can be applied.We then construct a set of linear,second-order and unconditionally energy stable schemes for the reformulated systems.These schemes lead to decoupled linear equations with constant coefficients at each time step so that,they can be implemented easily and very effieiepntly.We present,ample numerical results to demonstrate the stability and accuracy of our SAV schemes.3.We consider gradient flows with disparate terms in the free energy that cannot be efficiently handled with the scalar auxiliary variable(SAV)approach,and we develop the multiple scalar auxiliary variable(MSAV)approach to deal with these cases.We apply the MSAV approach to the phase-field vesicle membrane(PF-VMEM)model which,in addition to some usual nonlinear terms in the free energy,has two additional penalty terms to enforce the volume and surface area.The MSAV approach enjoys the same comput,ational advantages as the SAV approach but can handle free energies with multiple disparate terms such as the volume and surface area constraints in the PF-VMEM model.The MSAV schemes are unconditional energy stable and second-order accurate in time and lead to decoupled elliptic equations with constant coefficients to solve at each time step.Hence,these schemes are easy to implement and extremely efficient when coupled with an adaptive time stepping.Ample numerical results are presented to validate the stability and accuracy of the MSAV schemes.4.We develop unconditionally energy stable numerical schemes for the vesicle to vesicle adhesion model by using the MSAV approach.We introduce three scalar auxiliary variables,and by treating the adhesion potential term explicitly,numerical schemes can be decoupled and efficiently solved.Finally we only need to solve six fourth order equations totally.Some numerical simulations have been presented to validate the accuracy and stability of the numerical schemes we constructed in 3D.