The purpose of the current article is to study the H^(1)-stability for all positive time of the linearly extrapolated BDF2 timestepping scheme for the magnetohydrodynamics and Boussinesq equations.Specifically,we disc...The purpose of the current article is to study the H^(1)-stability for all positive time of the linearly extrapolated BDF2 timestepping scheme for the magnetohydrodynamics and Boussinesq equations.Specifically,we discretize in time using the linearly backward differentiation formula,and by employing both the discrete Gronwall lemma and the discrete uniform Gronwall lemma,we establish that each numerical scheme is uniformly bounded in the H^(1)-norm.展开更多
本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程...本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程,然后利用直接间断有限元求解耦合方程,最后使用BDF2方法,对时间格式进行离散。本文给出了详细的数值算法,并通过一个一维算例进行数值试验,验证了算法的有效性和收敛性。This paper mainly studies the effective numerical algorithm for the one-dimensional extended Fisher-Kolmogorov equation. By combining the BDF2 time discretization format with the direct discontinuous finite element algorithm, the one-dimensional extended Fisher-Kolmogorov equation is solved. Firstly, an auxiliary variable is introduced to transform the fourth-order extended Fisher-Kolmogorov equation into a low-order coupled equation. Then, the coupled equation is solved by using the direct discontinuous finite element method. Finally, the BDF2 method is used to discretize the time scheme. The detailed numerical algorithm is presented in this paper, and a one-dimensional example is used for numerical experiments to verify the effectiveness and convergence of the algorithm.展开更多
In this paper,a BDF2 modular grad-div algorithm for the Stokes/Darcy model is constructed.This method not only effectively avoids solver breakdown,but also increases computational efficiency for increasing parameter v...In this paper,a BDF2 modular grad-div algorithm for the Stokes/Darcy model is constructed.This method not only effectively avoids solver breakdown,but also increases computational efficiency for increasing parameter values.Herein,complete stability and error analysis are provided.Finally,some numerical tests are proposed to justify the theoretical analysis.展开更多
In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-or...In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.展开更多
针对非线性Benjamin-Bona-Mahony (BBM)方程,在时间上构造了2阶的Backward differential formula (BDF2)时间离散格式,在空间上采用双线性单元和零阶RT单元的混合有限元方法,研究了其超收敛性质.首先,利用变换技巧给出关于逼近方程的稳...针对非线性Benjamin-Bona-Mahony (BBM)方程,在时间上构造了2阶的Backward differential formula (BDF2)时间离散格式,在空间上采用双线性单元和零阶RT单元的混合有限元方法,研究了其超收敛性质.首先,利用变换技巧给出关于逼近方程的稳定性.其次,利用逼近解的有界性得到关于其原始变量u的一个超逼近结果,进而得到其中间变量q的超逼近结果.最后利用一个算例验证理论结果的正确性.展开更多
The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial ...The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.展开更多
An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectr...An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectral method is applied for the spatial approximation.The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model,primarily due to the introduction of the fractional Laplacian.This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian.Subsequently,the unique solvability and the modified energy dissipation law are theoretically guaranteed.We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term.Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian.In particular,we demonstrate that the adaptive time-stepping strategy,compared with the uniform time steps,captures the multiple time scale evolutions of the solution in simulations.展开更多
本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合...本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合适的数值通量,确保离散格式的稳定性。最后,针对时间导数项应用L1近似公式与BDF2时间差分的离散方法,建立全离散DDG格式。文中详细给出数值格式的构造过程并严格证明该算法的稳定性。数值实验部分选取无已知解析解的水波模型,验证该算法在时空离散上的高精度特性。This paper numerically solves the nonlocal viscous water wave model based on the Direct Discontinuous Galerkin (DDG) method. This algorithm combines the L1 approximation formula with the BDF2 method to systematically construct an efficient numerical algorithm for nonlinear time-fractional partial differential equations. Firstly, the integration by parts method is used to reduce the order of the weak formula. Secondly, the stability of the discrete scheme is ensured by introducing the boundary term and constructing a stable numerical flux. Finally, for the time derivative term, the discrete methods of the L1 approximation formula and the BDF2 time difference are applied to construct the fully discrete DDG scheme. This paper gives a detailed description of the construction process of the numerical scheme and strictly proves the stability of this algorithm. In the numerical experiment part, a water wave model without a known analytical solution is selected to verify the high-precision characteristics of this algorithm in space-time discretization.展开更多
文摘The purpose of the current article is to study the H^(1)-stability for all positive time of the linearly extrapolated BDF2 timestepping scheme for the magnetohydrodynamics and Boussinesq equations.Specifically,we discretize in time using the linearly backward differentiation formula,and by employing both the discrete Gronwall lemma and the discrete uniform Gronwall lemma,we establish that each numerical scheme is uniformly bounded in the H^(1)-norm.
文摘本文主要研究一维扩展的Fisher-Kolmogorov方程的有效数值算法。通过结合BDF2时间离散格式与直接间断有限元算法对一维扩展的Fisher-Kolmogorov方程进行求解。首先,引入辅助变量,将四阶的扩展的Fisher-Kolmogorov方程转化为低阶耦合方程,然后利用直接间断有限元求解耦合方程,最后使用BDF2方法,对时间格式进行离散。本文给出了详细的数值算法,并通过一个一维算例进行数值试验,验证了算法的有效性和收敛性。This paper mainly studies the effective numerical algorithm for the one-dimensional extended Fisher-Kolmogorov equation. By combining the BDF2 time discretization format with the direct discontinuous finite element algorithm, the one-dimensional extended Fisher-Kolmogorov equation is solved. Firstly, an auxiliary variable is introduced to transform the fourth-order extended Fisher-Kolmogorov equation into a low-order coupled equation. Then, the coupled equation is solved by using the direct discontinuous finite element method. Finally, the BDF2 method is used to discretize the time scheme. The detailed numerical algorithm is presented in this paper, and a one-dimensional example is used for numerical experiments to verify the effectiveness and convergence of the algorithm.
基金Supported by NSFC(12171376,2020-JCJQ-ZD-029)Natural Science Foundation of Hubei Province(2019CFA007)the Fundamental Research Funds for the Central Universities(2042021kf0050)。
基金Supported by the Provincial Natural Science Foundation of Shanxi(201901D111123)Key Research and Development(R&D)Projects of Shanxi Province(201903D121038).
文摘In this paper,a BDF2 modular grad-div algorithm for the Stokes/Darcy model is constructed.This method not only effectively avoids solver breakdown,but also increases computational efficiency for increasing parameter values.Herein,complete stability and error analysis are provided.Finally,some numerical tests are proposed to justify the theoretical analysis.
基金Natural Science Foundation of Hubei Province(2019CFA007)Supported by NSFC(11771035).
文摘In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.
文摘针对非线性Benjamin-Bona-Mahony (BBM)方程,在时间上构造了2阶的Backward differential formula (BDF2)时间离散格式,在空间上采用双线性单元和零阶RT单元的混合有限元方法,研究了其超收敛性质.首先,利用变换技巧给出关于逼近方程的稳定性.其次,利用逼近解的有界性得到关于其原始变量u的一个超逼近结果,进而得到其中间变量q的超逼近结果.最后利用一个算例验证理论结果的正确性.
文摘The second-order backward differential formula(BDF2)and the scalar auxiliary variable(SAV)approach are applied to con‐struct the linearly energy stable numerical scheme with the variable time steps for the epitaxial thin film growth models.Under the stepratio condition 0<τ_(n)/τ_(n-1)<4.864,the modified energy dissipation law is proven at the discrete levels with regardless of time step size.Nu‐merical experiments are presented to demonstrate the accuracy and efficiency of the proposed numerical scheme.
基金support by the National Natural Science Foundation of China(Nos.11701081,11861060)the State Key Program of National Natural Science Foundation of China(No.61833005)ZhiShan Youth Scholar Program of SEU.
文摘An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^(-α),α∈(0,1).The Fourier pseudo-spectral method is applied for the spatial approximation.The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model,primarily due to the introduction of the fractional Laplacian.This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian.Subsequently,the unique solvability and the modified energy dissipation law are theoretically guaranteed.We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term.Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian.In particular,we demonstrate that the adaptive time-stepping strategy,compared with the uniform time steps,captures the multiple time scale evolutions of the solution in simulations.
文摘本文基于直接间断有限元(DDG)方法数值求解非局部粘性水波模型。该算法结合L1近似公式与BDF2方法,系统构建了非线性时间分数阶偏微分方程的高效数值算法。首先,运用分部积分法对模型的弱形式进行降阶处理。其次,通过引入边界项和选用合适的数值通量,确保离散格式的稳定性。最后,针对时间导数项应用L1近似公式与BDF2时间差分的离散方法,建立全离散DDG格式。文中详细给出数值格式的构造过程并严格证明该算法的稳定性。数值实验部分选取无已知解析解的水波模型,验证该算法在时空离散上的高精度特性。This paper numerically solves the nonlocal viscous water wave model based on the Direct Discontinuous Galerkin (DDG) method. This algorithm combines the L1 approximation formula with the BDF2 method to systematically construct an efficient numerical algorithm for nonlinear time-fractional partial differential equations. Firstly, the integration by parts method is used to reduce the order of the weak formula. Secondly, the stability of the discrete scheme is ensured by introducing the boundary term and constructing a stable numerical flux. Finally, for the time derivative term, the discrete methods of the L1 approximation formula and the BDF2 time difference are applied to construct the fully discrete DDG scheme. This paper gives a detailed description of the construction process of the numerical scheme and strictly proves the stability of this algorithm. In the numerical experiment part, a water wave model without a known analytical solution is selected to verify the high-precision characteristics of this algorithm in space-time discretization.
