In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial deriv...In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.展开更多
We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equati...We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.展开更多
Based on the finite difference discretization of partial differential equations, we propose a kind of semi-implicit θ-schemes of incremental unknowns type for the heat equation with time-dependent coefficients. The s...Based on the finite difference discretization of partial differential equations, we propose a kind of semi-implicit θ-schemes of incremental unknowns type for the heat equation with time-dependent coefficients. The stability of the new schemes is carefully studied. Some new types of conditions give better stability when θ is closed to 1/2 even if we have variable coefficients.展开更多
In fact,the popular semi-implicit time difference scheme of spectral model still includes some important linear terms using time explicit difference scheme,and the major terms are directly related to fast internal-and...In fact,the popular semi-implicit time difference scheme of spectral model still includes some important linear terms using time explicit difference scheme,and the major terms are directly related to fast internal-and external-gravity waves in the atmospheric forecasting equation. Additionally,due to using time difference on two terms at different time.the popular scheme artificially introduces unbalance between pressure gradient force and Coriolis force terms while numerically computing their small difference between large quantities.According to the computational stability analysis conducted to the linear term time difference scheme in simple harmonic motion equation,one improved semi-implicit time difference scheme is also designed in our study.By adopting a kind of revised time-explicit-difference scheme to these linear terms that still included in spectral model governing equations,the defect of spectral model which only partly using semi-implicit integrating scheme can be overcome effectively.Moreover,besides all spectral coefficients of prognostic equations,especially of Helmholtz divergence equation,can be worked out without any numerical iteration,the time-step (computation stability) can also be enlarged (enhanced) by properly introducing an adjustable coefficient.展开更多
We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions.We employ the standard semi-implicit numerical scheme,which treats the linear fourth-order dissipatio...We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions.We employ the standard semi-implicit numerical scheme,which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly.Under natural constraints on the time step we prove strict phase separation and energy stability of the semiimplicit scheme.This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.展开更多
文摘In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.
文摘We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.
基金This project is partially supported by Natural Science Foundation of Gansu Province under Grant 3ZS041-A25-011 by National Natural Science Foundation under Grant 10471056.
文摘Based on the finite difference discretization of partial differential equations, we propose a kind of semi-implicit θ-schemes of incremental unknowns type for the heat equation with time-dependent coefficients. The stability of the new schemes is carefully studied. Some new types of conditions give better stability when θ is closed to 1/2 even if we have variable coefficients.
基金The project is supported by the Beijing New Star Program of Science and Technology of China during 2001-2004 under Grant No.H013610330119.
文摘In fact,the popular semi-implicit time difference scheme of spectral model still includes some important linear terms using time explicit difference scheme,and the major terms are directly related to fast internal-and external-gravity waves in the atmospheric forecasting equation. Additionally,due to using time difference on two terms at different time.the popular scheme artificially introduces unbalance between pressure gradient force and Coriolis force terms while numerically computing their small difference between large quantities.According to the computational stability analysis conducted to the linear term time difference scheme in simple harmonic motion equation,one improved semi-implicit time difference scheme is also designed in our study.By adopting a kind of revised time-explicit-difference scheme to these linear terms that still included in spectral model governing equations,the defect of spectral model which only partly using semi-implicit integrating scheme can be overcome effectively.Moreover,besides all spectral coefficients of prognostic equations,especially of Helmholtz divergence equation,can be worked out without any numerical iteration,the time-step (computation stability) can also be enlarged (enhanced) by properly introducing an adjustable coefficient.
基金supported in part by Hong Kong RGC grant GRF Nos.16307317,16309518partially supported by the NSFC grants Nos.11731006,K20911001,NSFC/RGC No.11961160718the Science Challenge Project(No.TZ2018001)。
文摘We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions.We employ the standard semi-implicit numerical scheme,which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly.Under natural constraints on the time step we prove strict phase separation and energy stability of the semiimplicit scheme.This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.
