Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work,we are concerned with the stability and convergence analysis of the second-order backward difference formula(BDF2)with variable steps for the molecular beam epitaxial model without slope selection.We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint.Then we show that it preserves an energy dissipation law if the adjacent time-step ratios satisfy r_(k):=τ_(k)/τ_(k-1)<3.561.Moreover,with a novel discrete orthogonal convolution kernels argument and some new estimates on the corresponding positive definite quadratic forms,the L^(2)norm stability and rigorous error estimates are established,under the same step-ratio constraint that ensures the energy stability,i.e.,0<r_(k)<3.561.This is known to be the best result in the literature.We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

SEMI-IMPLICIT SPECTRAL DEFERRED CORRECTION METHODS BASED ON SECOND-ORDER TIME INTEGRATION SCHEMES FOR NONLINEAR PDES

In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.

Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation

In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization,and by applying the Fourier spectral collocation to the spatial discretization.In addition,two stabilization terms in different forms are added for the sake of the numerical stability.We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme,combined with the rough error estimate and the refined estimate.By regarding the numerical solution as a small perturbation of the exact solution,we are able to justify the discrete?^(∞)bound of the numerical solution,as a result of the rough error estimate.Subsequently,the refined error estimate is derived to obtain the optimal rate of convergence,following the established?∞bound of the numerical solution.Moreover,the energy stability is also rigorously proved with respect to a modified energy.The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work,and the energy stability estimate has greatly improved the corresponding result therein.

Second-order two-scale analysis and numerical algorithms for the hyperbolic–parabolic equations with rapidly oscillating coefficients

We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.

BBM-Burgers方程的非协调有限元方法的超收敛分析

研究Benjamin-Bona-Mahony-Burgers(BBM-Burgers)方程的非协调EQ_(1)^(rot)元的线性化BDF格式下的超收敛性质。通过使用数学归纳法来处理非线性项,并利用该单元已有的高精度结果及插值后处理技术,得到了在对空间剖分尺度和时间步长无网格比约束的前提下,关于离散H^(1)-模意义下具有O(h^(2)+τ^(2))阶的超逼近和超收敛结果。最后,通过给出数值算例验证了理论分析的正确性。

二维分数阶发展型方程的正式的二阶BDF交替方向隐式紧致差分格式

该文将研究二维分数阶发展型方程的正式的二阶向后微分公式(BDF)的交替方向隐式(ADI)紧致差分格式.在时间方向上用二阶向后微分公式离散一阶时间导数,积分项用二阶卷积求积公式近似,在空间方向上用四阶精度的紧致差分离散二阶空间导数得到全离散紧致差分格式.基于与卷积求积相对应的实二次型的非负性,利用能量方法研究了差分格式的稳定性和收敛性,理论结果表明紧致差分格式的收敛阶为O(k^(a+1)+h_1~4+h_2~4),其中k为时间步长,h_1和h_2分别是空间x和y方向的步长.最后,数值算例验证了理论分析的正确性.

A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.

UNCONDITIONALLY OPTIMAL ERROR ANALYSIS OF THE SECOND-ORDER BDF FINITE ELEMENT METHOD FOR THE KURAMOTO-TSUZUKI EQUATION

This paper aims to study a second-order semi-implicit BDF finite element scheme for the Kuramoto-Tsuzuki equations in two dimensional and three dimensional spaces.The proposed scheme is stable and the nonlinear term is linearized by the extrapolation technique.Moreover,we prove that the error estimate in L^(2)-norm is unconditionally optimal which means that there has not any restriction on the time step and the mesh size.Finally,numerical results are displayed to illustrate our theoretical analysis.

非线性偏积分微分方程紧隐显BDF方法的误差分析

本文研究一类非线性抛物型偏积分微分方程数值格式的误差分析,其格式在空间方向采用紧致差分方法,时间方向采用隐显BDF方法,积分项利用梯形数值求积公式进行求解,给出了算法的收敛阶O(τ^2+h^4);通过数值算例验证了数值格式的准确性和有效性。

Optimal Error Estimates of the Crank-Nicolson Scheme for Solving a Kind of Decoupled FBSDEs

In this paper, under weak conditions, we theoretically prove the second-order convergence rate of the Crank-Nicolson scheme for solving a kind of decoupled forward-backward stochastic differential equations.

Landau-Lifshitz方程基于外推法的二阶BDF投影方法

研究了求解Landau-Lifshitz方程的二阶BDF投影有限元算法,使得数值解可逐点满足单位长度的非凸约束.借助数学归纳法,得到了精确解和有限元解的最优L^(2)误差估计.

Second-order schemes for solving decoupled forward backward stochastic differential equations

In this paper,by using trapezoidal rule and the integration-by-parts formula of Malliavin calculus,we propose three new numerical schemes for solving decoupled forward-backward stochastic differential equations.We theoretically prove that the schemes have second-order convergence rate.To demonstrate the effectiveness and the second-order convergence rate,numerical tests are given.

A Second-Order Semi-Implicit Method for the Inertial Landau-Lifshitz-Gilbert Equation

Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at nanosecond timescales,the results of the iLLG model are in nice agreements with those of the LLG model,in consistency with the parabolic feature of the iLLG model.

ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.

BOX-LINE RELAXATION SCHEMES FOR SOLVING THE STEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS USING SECOND-ORDER UPWIND DIFFERENCING

We extend the SCGS smoothing procedure (Symmetrical Collective Gauss-Seidel relaxation) proposed by S. P. Vanka[4], for multigrid solvers of the steady viscous incompressible Navier-Stokes equations, to corresponding line-wise versions. The resulting relaxation schemes are integrated into the multigrid solver based on second-order upwind differencing presented in [5]. Numerical comparisons on the efficiency of point-wise and line-wise relaxations are presented

二维瞬态热传导问题的无单元Galerkin法分析

采用无单元Galerkin(element-free Galerkin,EFG)法求解具有混合边界条件的二维瞬态热传导问题.首先采用二阶向后微分公式离散热传导方程的时间变量,将该问题转化为与时间无关的混合边值问题;然后采用罚函数法处理Dirichlet边界条件,建立了二维瞬态热传导问题的无单元Galerkin法;最后基于移动最小二乘近似的误差结果,详细推导了无单元Galerkin法求解二维瞬态热传导问题的误差估计公式.给出的数值算例表明计算结果与解析解或已有数值解吻合较好,该方法具有较高的计算精度和较好的收敛性.

DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

This is one of our series works on discrete energy analysis of the variable-step BDF schemes.In this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM J.Numer.Anal.,58:2294-2314]and[Math.Comp.,90:1207-1226]for our previous works on the BDF2 scheme.To this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation law.Mesh-robust stability and convergence analysis in the L^(2) norm are then obtained.Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios.We also present numerical tests to support our theoretical results.

Symmetric Energy-Conserved Splitting FDTD Scheme for the Maxwell’s Equations

In this paper,a new symmetric energy-conserved splitting FDTD scheme(symmetric EC-S-FDTD)for Maxwell’s equations is proposed.The new algorithm inherits the same properties of our previous EC-S-FDTDI and EC-S-FDTDII algorithms:energy-conservation,unconditional stability and computational efficiency.It keeps the same computational complexity as the EC-S-FDTDI scheme and is of second-order accuracy in both time and space as the EC-S-FDTDII scheme.The convergence and error estimate of the symmetric EC-S-FDTD scheme are proved rigorously by the energy method and are confirmed by numerical experiments.