The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) pro...The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.展开更多
In this paper, a predictor-corrector finite difference-streamline diffusion finite element scheme is constructed for time-dependent quasilinear convection-diffusion problem. For the scheme considered, the solvility is...In this paper, a predictor-corrector finite difference-streamline diffusion finite element scheme is constructed for time-dependent quasilinear convection-diffusion problem. For the scheme considered, the solvility is proved, and the error estimate in L(L2)-norm is established. It has 2-order accuracy in time direction and quasi-optimal order accuracy in space variables.展开更多
For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fract...For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fractional steps techniques are needed to convert a multi-dimensional problem into a series of successive one-dimensional problems. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources and predicting the consequences of seawater intrusion and protection projects.展开更多
A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutio...A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutions is obtained in L2 under milder restrictions in the temporal stepsize and spatial stepsize than those required in [1].展开更多
文摘The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.
文摘In this paper, a predictor-corrector finite difference-streamline diffusion finite element scheme is constructed for time-dependent quasilinear convection-diffusion problem. For the scheme considered, the solvility is proved, and the error estimate in L(L2)-norm is established. It has 2-order accuracy in time direction and quasi-optimal order accuracy in space variables.
基金Project supported by the Major State Basic Research Program of China (No.G1999032803)the National Tackling Key Problems Program (No.20050200069)the National Natural Science Foundation of China (Nos.10372052, 10271066)the Doctoral Foundation of Ministry of Education of China (No.20030422047).
文摘For the three-dimensional convection-dominated problem of dynamics of fluids in porous media, the second order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Fractional steps techniques are needed to convert a multi-dimensional problem into a series of successive one-dimensional problems. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates are adopted. Optimal order estimates are derived to determine the error in the second order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources and predicting the consequences of seawater intrusion and protection projects.
文摘A singlestep characteristic finite difference method is given based on the linear and quadratic interpolations for solving two-dimensional nonlinear convection-diffusionproblems. The convergence of approximate solutions is obtained in L2 under milder restrictions in the temporal stepsize and spatial stepsize than those required in [1].
