A local energy-preserving scheme for Zakharov system

In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local time- space region which is independent of the boundary condition and more essential than the global energy conservation law. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving （LEP） scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.

Spline Solution for the Nonlinear Schrödinger Equation

We develop an exponential spline interpolation method to solve the nonlinear Schr&oumldinger equation. The truncation error and stability analysis of the method are investigated and the method is shown to be unconditionally stable. The conservation quantities are computed to determine the conservation properties of the problem. We will describe the method and present numerical tests by two problems. The numerical simulations results demonstrate the well performance of the proposed method.

An Upwind Mixed Finite Volume Element-fractional Step Method and Convergence Analysis for Three-dimensional Compressible Contamination Treatment from Nuclear Waste

In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in L2norm. Finally, numerical examples show the effectiveness and practicability for some actual problems.

A Generalized Numerical Approach for Modeling Multiphase Flow and Transport in Fractured Porous Media

A physically based numerical approach is presented for modeling multiphase flow and transport processes in fractured rock.In particular,a general framework model is discussed for dealing with fracture-matrix interactions,which is applicable to both continuum and discrete fracture conceptualization.The numerical modeling approach is based on a general multiple-continuum concept,suitable for modeling any types of fractured reservoirs,including double-,triple-,and other multiplecontinuum conceptual models.In addition,a new,physically correct numerical scheme is discussed to calculate multiphase flow between fractures and the matrix,using continuity of capillary pressure at the fracture-matrix interface.The proposed general modeling methodology is verified in special cases using analytical solutions and laboratory experimental data,and demonstrated for its application in modeling flow through fractured vuggy reservoirs.

ON THE VERTICAL DISCRETIZATION OF ATMOSPHERIC MODELS—PART Ⅱ:VERTICAL DISCRETIZATION

Two schemes for vertical discretization of the model are proposed,one with equal △lnσ and the other in terms of Tschebyscheff polynomials.It is proved that in adiabatic and inviscid cases if the meteorological elements and related physical quantities are continuous in time and in the horizontal,the total energy and total mass are conserved within a high approximation respectively, and there is a correct conversion between total kinetic and total potential energy.Numerical computations show that the schemes both have high accuracy.For example,in integrating the hydrostatic equation the computational errors of geopotential height resulting from the schemes are much less than those resulting from EC79 in a-coordinate.

An Upwind Mixed Volume Element-Fractional Step Method on a Changing Mesh for Compressible Contamination Treatment from Nuclear Waste

In this paper the authors discuss the numerical simulation problem of threedimensional compressible contamination treatment from nuclear waste.The mathematical model is defined by an initial-boundary nonlinear convection-diffusion system of four partial differential equations:a parabolic equation for the pressure,two convection-diffusion equations for the concentrations of brine and radionuclide and a heat conduction equation for the temperature.The pressure appears within the concentration equations and heat conduction equation,and the Darcy velocity controls the concentrations and the temperature.The pressure is solved by the conservative mixed volume element method,and the order of the accuracy is improved by the Darcy velocity.The concentration of brine and temperature are computed by the upwind mixed volume element method on a changing mesh,where the diffusion is discretized by a mixed volume element and the convection is treated by an upwind scheme.The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy.The mixed volume element has the local conservation of mass and energy,and it can obtain the brine and temperature and their adjoint vector functions simultaneously.The conservation nature plays an important role in numerical simulation of underground fluid.The concentrations of radionuclide factors are solved by the method of upwind fractional step difference and the computational work is decreased by decomposing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup.By the theory and technique of a priori estimates of differential equations,we derive an optimal order result in L^(2) norm.Numerical examples are given to show the effectiveness and practicability and the composite method is testified as a powerful tool to solve the well-known actual problem.