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
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
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 firs...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.展开更多
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 linea...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.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.
基金supported by National Natural Science Foundation of China(Grant No.12071216)supported by National Natural Science Foundation of China(Grant No.11731006)+2 种基金the NNW2018-ZT4A06 projectsupported by National Natural Science Foundation of China(Grant Nos.11822111,11688101 and 11731006)the Science Challenge Project(Grant No.TZ2018001)。
文摘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.
基金supported by NSF of China under grant number 12071216supported by NNW2018-ZT4A06 project+1 种基金supported by NSF of China under grant numbers 12288201youth innovation promotion association(CAS).
文摘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.
