THE UPWIND FINITE DIFFERENCE FRACTIONAL STEPS METHOD FOR NONLINEAR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, trod two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.

The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.

FULL DISCRETE NONLINEAR GALERKIN METHOD FOR THE NAVIER-STOKES EQUATIONS

This paper deals with the inertial manifold and the approximate inertialmanifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore,we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.

ON THE ERROR ESTIMATES OF A NEW OPERATE SPLITTING METHOD FOR THE NAVIER-STOKES EQUATIONS

In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.

AViscosity-Splitting Method for the Navier-Stokes/ Darcy Problem

In this report,we give a viscosity splitting method for the Navier-Stokes/Darcy problem.In this method,the Navier-Stokes/Darcy equation is solved in three steps.In the first step,an explicit/implicit formulation is used to solve the nonlinear problem.We introduce an artificial diffusion term qDu in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability,especially for small viscosity parameter n,by choosing suitable parameter q.In the second step,we solve the Stokes equation for velocity and pressure.In the third step,we solve the Darcy equation for the piezometric head in the porous media domain.We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy’s equation.The stability analysis,under some condition △t≤k0,k0>0,is given.The error estimates prove our method has an optimal convergence rates.Finally,some numerical results are presented to show the performance of our algorithm.

High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

Within the projection schemes for the incompressible Navier-Stokes equations(namely"pressure-correction"method),we consider the simplest method(of order one in time)which takes into account the pressure in both steps of the splitting scheme.For this scheme,we construct,analyze and implement a new high order compact spatial approximation on nonstaggered grids.This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions(without error on the velocity)which could be extended to more general splitting.We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis.Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions.Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations(including the driven cavity benchmark)to illustrate the theoretical results.

Fast flow simulation study of pulsating ventilation performance on air contaminant removal

Fast flow simulation is imperative in the design of pulsating ventilation,which is potentially efficient in indoor air contaminant removal.The execution of the conventional CFD method requires considerable amount of computational resources.In this study,five different numerical schemes were proposed based on fast fluid dynamics(FFD)and fractional step(FS)methods,and were evaluated to achieve quick simulation of airflow/contaminant dispersion.One of these numerical schemes was identified with the best overall computing efficiency for investigating the performance of pulsating ventilation.With this numerical scheme at hand,the air contaminant removal effectiveness of different ventilation types was evaluated.Two kinds of pulsating ventilation and one kind of steady ventilation were tested upon a benchmark isothermal mixing chamber.The effect of adjusting supply velocity parameters on the ventilation performance was also investigated.CO_(2)concentration,airflow pattern,and vortex structure of different ventilation types were illustrated and analyzed.The results reveal that the FS method is more suitable for transient simulation of wall-bounded indoor airflow than the FFD method,and 34%–51%of computing time could be saved compared to the conventional CFD method.Regarding the choice of ventilation type,steady ventilation might result in short-circuit airflow and stagnant zones;alternatively,pulsating ventilation has greater potential in air contaminant removal due to its ever-changing vortex structure.

AN OPERATOR-SPLITTING ALGORITHM FOR ADVECTION-DIFFUSION-REACTION EQUATION

An operator-splitting algorithm for three-dimensional advection-diffusion-reaction equation is presented.The method of characteristics is adopted for the pure advection operator, the explicit difference scheme is used for diffusion,and a prediction-correction scheme is em- ployed for reaction.The condition for stability of the algorithm is analysed.Severall inear and nonlinear examples are illustrated to test the convergence and accuracy of the numerical proce- dure,and satisfactory agreements between computed and analytical solutions are achieved.Due to its simplicity,stability,and validity for both one-and two-dimensional problems,the success- ful algorithm can be used to numerical simulations of viscous fluid flows,the transport of pollu- tants and sedimentations in reservoirs,lakes,rivers,estuaries and other environments,cooling- problems in heat or nuclear power plants,etc.

Mathematical Analysis of the Jin-Neelin Model of El Niño-Southern-Oscillation

The Jin-Neelin model for the El Nio–Southern Oscillation(ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature(SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.

CONVERGENCE OF CHORIN-MARSDEN FOR MULA FOR THENAVIER-STOKES EQUATIONS ON CONVEX DOMAINS

We prove the convergence of the Chorin-Marsden product formula for solving the initial-boundary value problems of the Navier-Stokes equations on convex domains. As a particular case we consider the case of the half plane.