When modeling wave propagation in infinite space, it is necessary to have stable absorbing boundaries to effectively eliminate spurious reflections from the truncation boundaries. The SH wave equations for Perfectly M...When modeling wave propagation in infinite space, it is necessary to have stable absorbing boundaries to effectively eliminate spurious reflections from the truncation boundaries. The SH wave equations for Perfectly Matched Layers （PML） are deduced and their Crank-Nicolson scheme are presented in this paper. We use the second-, sixth-, and tenth-order finite difference and pseudo-spectral algorithms to compute the spatial derivatives. Two numerical models, a homogeneous isotropic medium and a multi-layer model with a cave, are designed to investigate how the absorbing boundary width and the algorithms determine PML effects. Numerical results show that, for PML, the low-order finite difference algorithms have fairly good absorbing effects when the absorbing boundary is thin, whereas, high-order algorithms always have good absorption when the boundary is thick. Finally, we discuss the reflection coefficient and point out its shortcomings, which is why we use the SNR to quantitatively scale the PML effects,展开更多
The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occu...The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occur and the general asymptotic behaviour is complex. This behaviour reflects for large times, even on compact sets, the complexity of the initial data at infinity.展开更多
We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping paramet...We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.展开更多
The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuo...The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuous Galerkin approximation is derived without using the nonclassical elliptic projection of the exact solution of the model problem. The global superconvergence of one order higher than the traditional error estimate is also obtained through the postprocessing technique.展开更多
This paper aims to look into the determination of effective area-average concentration and dispersion coefficient associated with unsteady flow through a small-diameter tube where a solute undergoes first-order chemic...This paper aims to look into the determination of effective area-average concentration and dispersion coefficient associated with unsteady flow through a small-diameter tube where a solute undergoes first-order chemical reaction both within the fluid and at the boundary. The reaction consists of a reversible component due to phase exchange between the flowing fluid and the wall layer, and an irreversible component due to absorption into the wall. To understand the dispersion, the governing equations along with the reactive boundary conditions are solved numerically using the Finite Difference Method. The resultant equation shows how the dispersion coefficient is influenced by the first-order chemical reaction. The effects of various dimensionless parameters e.g. Da （the Damkohler number）, a （phase partitioning number） and F （dimensionless absorption number） on dispersion are discussed. One of the results exposes that the dispersion coefficient may approach its steady-state limit in a short time at a high value of Damkohler number （say Da 〉 10） and a small but nonzero value of absorption rate （say P 〈0.5）.展开更多
The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of th...The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of the boundary.The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler's scheme in time and finite elements in space.The convergence of this discretization leads to the well-posedness of the problem.展开更多
基金supported jointly by the 973 Program (Grant No.2007CB209505)the National Natural Science Fund (Grant No.40704019,40674061)+1 种基金the School Basic Research Fund of Tsinghua University (JC2007030)PetroChina Innovation Fund (Grant No.060511-1-1)
文摘When modeling wave propagation in infinite space, it is necessary to have stable absorbing boundaries to effectively eliminate spurious reflections from the truncation boundaries. The SH wave equations for Perfectly Matched Layers （PML） are deduced and their Crank-Nicolson scheme are presented in this paper. We use the second-, sixth-, and tenth-order finite difference and pseudo-spectral algorithms to compute the spatial derivatives. Two numerical models, a homogeneous isotropic medium and a multi-layer model with a cave, are designed to investigate how the absorbing boundary width and the algorithms determine PML effects. Numerical results show that, for PML, the low-order finite difference algorithms have fairly good absorbing effects when the absorbing boundary is thin, whereas, high-order algorithms always have good absorption when the boundary is thick. Finally, we discuss the reflection coefficient and point out its shortcomings, which is why we use the SNR to quantitatively scale the PML effects,
文摘The authors study the asymptotic behaviour of solutions of the heat equation and a number of evolution equations using scaling techniques. It is proved that in the framework of bounded data stabilization need not occur and the general asymptotic behaviour is complex. This behaviour reflects for large times, even on compact sets, the complexity of the initial data at infinity.
基金supported by Australian Research Council Discovery Project (Grant No. DP170101060)National Natural Science Foundation of China (Grant No. 11201498)the China Scholarship Council (Grant No. 201606495010)
文摘We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.
基金supported by the National Natural Science Foundation of China under Grant Nos.10671184 and 10971203
文摘The standard finite elements of degree p over the rectangular meshes are applied to solve a kind of nonlinear viscoelastic wave equations with nonlinear boundary conditions, and the superclose property of the continuous Galerkin approximation is derived without using the nonclassical elliptic projection of the exact solution of the model problem. The global superconvergence of one order higher than the traditional error estimate is also obtained through the postprocessing technique.
文摘This paper aims to look into the determination of effective area-average concentration and dispersion coefficient associated with unsteady flow through a small-diameter tube where a solute undergoes first-order chemical reaction both within the fluid and at the boundary. The reaction consists of a reversible component due to phase exchange between the flowing fluid and the wall layer, and an irreversible component due to absorption into the wall. To understand the dispersion, the governing equations along with the reactive boundary conditions are solved numerically using the Finite Difference Method. The resultant equation shows how the dispersion coefficient is influenced by the first-order chemical reaction. The effects of various dimensionless parameters e.g. Da （the Damkohler number）, a （phase partitioning number） and F （dimensionless absorption number） on dispersion are discussed. One of the results exposes that the dispersion coefficient may approach its steady-state limit in a short time at a high value of Damkohler number （say Da 〉 10） and a small but nonzero value of absorption rate （say P 〈0.5）.
文摘The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of the boundary.The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler's scheme in time and finite elements in space.The convergence of this discretization leads to the well-posedness of the problem.
