In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we i...In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.展开更多
In this work,we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form.The L1 implicit scheme is shown to preserve a variational energy dissipation law o...In this work,we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form.The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools,i.e.,the discrete orthogonal convolution kernels and discrete complementary convolution kernels.Then the discrete embedding techniques and the fractional Gronwall inequality are applied to establish an L^(2)norm error estimate on nonuniform time meshes.An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.展开更多
As a promising strategy to adjust the order in the variable-order BDF algorithm,a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection.The temporal second-...As a promising strategy to adjust the order in the variable-order BDF algorithm,a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection.The temporal second-order convergence in the L^(2)norm is established under a convergence-solvability-stability(CSS)-consistent time-step constraint.The CSS-consistent condition means that the maximum stepsize limit required for convergence is of the same order to that for solvability and stability(in certain norms)as the small interface parameterε→0^(+).Similar to the backward Euler scheme,the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels,including the volume conservation,the energy dissipation law and L^(2)norm boundedness.Numerical tests are included to support the theoretical results.展开更多
基金supported by National Science Foundation of China(11071162)Shanghai Municipal Natural Science Foundation (09ZR1413500)
文摘In this paper, we will introduce how to apply Green＇s function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen＇s principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green＇s function of the linearized system and micro-local analysis, such as frequency decomposition and so on.
基金The authors would like to thank Dr.Bingquan Ji for his help on numerical computations.H.-L.Liao is supported by the National Natural Science Foundation of China(Grant 12071216)J.Wang is supported by the Hunan Provincial Innovation Foundation for Postgraduate(Grant XDCX2020B078).
文摘In this work,we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form.The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools,i.e.,the discrete orthogonal convolution kernels and discrete complementary convolution kernels.Then the discrete embedding techniques and the fractional Gronwall inequality are applied to establish an L^(2)norm error estimate on nonuniform time meshes.An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.
基金The work of H.-L.Liao is supported by NSF of China(Grant No.12071216)。
文摘As a promising strategy to adjust the order in the variable-order BDF algorithm,a time filtered backward Euler scheme is investigated for the molecular beam epitaxial equation with slope selection.The temporal second-order convergence in the L^(2)norm is established under a convergence-solvability-stability(CSS)-consistent time-step constraint.The CSS-consistent condition means that the maximum stepsize limit required for convergence is of the same order to that for solvability and stability(in certain norms)as the small interface parameterε→0^(+).Similar to the backward Euler scheme,the time filtered backward Euler scheme preserves some physical properties of the original problem at the discrete levels,including the volume conservation,the energy dissipation law and L^(2)norm boundedness.Numerical tests are included to support the theoretical results.
