Analysis Of Time-Efficient Algorithms For Swift-Hohenberg Equations

The Swift-Hohenberg equation is a kind of phase field model with energy dissipation proper-ties,which has been widely applied in physics,chemistry,materials science,laser physics and many other non-linear fields.In general,the solution may undergo large variations initially or at some time,while the later dynamic coarsening evolves rather slowly until it reaches a steady state,which implies the temporal evolution of Swift-Hohenberg equation involving multiple time scales.The mul-tiscale behavior indicates that the large uniform time steps may bring about miscalculation evidently.Under the circumstances,adaptive time-stepping approximations enable substantial computational efficiency improvements while guaranteeing the numerical accuracy.In this paper,adaptive time-stepping algorithms are put forward for multiscale properties,both for the classical and the time-fractional Swift-Hohenberg equation.The unique solvability,energy stability and convergence of the proposed schemes are analyzed rigorously.In chapter 2,an implicit variable steps BDF2 schemes for the classical Swift-Hohenberg equation is considered.The global second order accuracy for the numerical scheme in L²norm is achieved.Ap-plying Brouwer’s fixed point theorem,unique solvability of the scheme is proved strictly.By rewrit-ing the BDF2 method in the discrete convolution form,together with the energy analysis method,the global energy stability of the scheme is proved.The boundedness of solution is obtained as the byprod-uct.Subsequently,by constructing the corresponding discrete orthogonal convolution（DOC）kernels for the BDF2 method and using the discrete Gr¨onwall inequality,the proposed scheme is proved to be convergent in L²norm.Some numerical examples are shown to illustrate the convergence accuracy,energy dissipation properties and the evolution of the solution.In chapter 3,an implicit uniform steps BDF3 scheme is established.The theoretical proof of the unique solvability and the energy stability is obtained correspondingly.We construct corresponding DOC kernels for the uniform BDF3 method.The global convergence order in the temporal direction is third order.Two-and three-dimensional numerical examples are used to illustrate the validity of the proposed scheme finally.In chapter 4,a fully discrete variable-stepping L1 implicit difference scheme for the time frac-tional Swift-Hohenberg equation is established.The unique solvability is demonstrated by Brouwer’s fixed point theorem.By constructing the corresponding DOC and discrete complementary con-volution（DCC）kernels for the L1 convolution kernel,the proposed scheme is shown to be glob-ally energy stable.Applying the properties of convolution kernels and the discrete fractional order Gr¨onwall inequality,the convergence analysis is carried out,which shows the temporal accuracy is min{τ^γα,τ^2-α}order,whereα∈（0,1）is the order of time-fractional derivatives,γ≥1 is a param-eter of the time grid.Finally,the validity of the above theoretical analysis is verified by numerical experiments,and the coarsening process is simulated for a long time.