The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],...The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.展开更多
We consider the multi-scale modeling of the isothermal chemical vapor infiltration (CVI) process for the fabrication of C/SiC composites. We first present a microscopic model in which the preform is regarded as a two-...We consider the multi-scale modeling of the isothermal chemical vapor infiltration (CVI) process for the fabrication of C/SiC composites. We first present a microscopic model in which the preform is regarded as a two-phase porous media describedby a dynamic pore-scale node-bond network during the fabrication process. We thendevelop a macroscopic model by a upscaling procedure based on the homogenizationtheory.展开更多
基金The work of Yue is supported in part by NSF of China under the grants No.11971342.
文摘The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.
基金The work of Yue is supported in part by NSF of China under the grant 10871190 and the National Basic Research Program under the Grant 2005CB321704The work of Zeng is supported in part by Flying Star Program of Northwestern Polytechnical University and NSF of China under the Grant 50802076.
文摘We consider the multi-scale modeling of the isothermal chemical vapor infiltration (CVI) process for the fabrication of C/SiC composites. We first present a microscopic model in which the preform is regarded as a two-phase porous media describedby a dynamic pore-scale node-bond network during the fabrication process. We thendevelop a macroscopic model by a upscaling procedure based on the homogenizationtheory.
