Efficient Numerical Method And Analysis Of Phase Field Model And Reaction-diffusion Equation Based On Adaptive BDF2

Time-adaptive is a strategy that automatically adjusts the time step size based on error es-timation to improve computational efficiency.It is mainly applied to handle time multi-scale problems and can significantly save computational time without sacrificing accuracy.The classical second-order backward differentiation formula（BDF2）is widely used for stiff prob-lems due to its strong stability.However,as a multi-step method,its theoretical analysis of time adaptive scheme is complex and challenging.This paper aims to investigate the stability and convergence analysis of the BDF2 scheme on a general time-nonuniform grid.A fully-discrete scheme satisfying the energy dissipation is constructed to ensure the thermodynamic consistency of the phase field model.To address the problem of overly strict limitations on the step ratio,the Discrete Complementary Convolution（DCC）kernel is introduced in this paper to expand the upper bound of the step ratio to 4.8645.To tackle the cost issue of nonlinear numerical iterative methods,an adaptive linear scheme based on the generalized Scalar Aux-iliary Variable（SAV）method is constructed.The application of the adaptive BDF2 scheme to nonconforming finite elements poses many challenges.In view of this,this paper presents a unified framework for corresponding convergence analysis based on the energy projection operator.Additionally,all the work in this paper is based on the mild step ratio limitation condition A1(i.e.,0≤r_k:=τ_k/τ_k-1≤r_max≈4.8645)for optimal error estimation.The main contributions of this paper are as follows:·Firstly,a time-adaptive fully implicit BDF2 scheme is proposed based on the Fourier pseudo-spectral method for solving the crystal phase-field model.We demonstrate that under the mild stepsize ratios condition A1,this scheme inherits the dissipation law of the modified energy,which numerically verifies the thermodynamic consistency of the crystal phase-field model.We establish the second-order optimal convergence analysis of the given scheme based on the discrete orthogonal convolution kernel,discrete com-plementary convolution kernel,and their extension properties under the condition A1.To the best of our knowledge,the required ratio condition A1 in this paper is the mildest among existing variable time-step BDF2 schemes for computing the crystal phase-field model.Numerical experiments are conducted to verify the theoretical correctness and algorithmic efficiency.·Secondly,an adaptive Implicit-Explicit BDF2 scheme for the Cahn-Hilliard equation is investigated on generalized SAV approach,based on the Fourier spectral method.We rigorously prove that the modified discrete energy satisfies the dissipation law uncondi-tionally.Under the condition A1,we establish a rigorous H¹-norm error estimate and achieve optimal second-order accuracy in time.The proof mainly relies on the discrete orthogonal convolution kernel and inequality zoom tools.It is worth noting that the pro-posed adaptive time-step scheme only requires solving a constant-coefficient linear sys-tem at each time level.In our analysis,the first-consistent BDF1 for the first step does not bring the order reduction in H¹-norm.The H¹bound of numerical solution under pe-riodic boundary conditions can be derived without any restriction（such as zero mean of the initial data）.Numerical experiments are consistent with the theoretical and analytical results,validating the correctness of the theory and the effectiveness of the algorithm.·Finally,an adaptive BDF2 fully discrete scheme is constructed based on the nonconform-ing finite element method for solving linear reaction-diffusion equations.By introducing a modified energy projection operator,a discrete Laplace operator,and a discrete orthogo-nal convolution kernel,we provide L²-norm and H¹-norm error estimates for the proposed scheme under mild restriction A1.In addition,using an improved discrete Gr（?）nwall in-equality and the combination of interpolation and projection operators,we obtain super-convergence results in the H¹-norm,and apply interpolation post-processing techniques to obtain global superconvergence results.Numerical experiments further validate the correctness of the theoretical analysis.