Discrete mass conservation and stability analysis of the ocean-atmosphere model with coupling conditions

In this work we considered bi-domain partial differential equations(PDEs)with two coupling interface conditions.The one domain is corresponding to the ocean and the second is to the atmosphere.The two coupling conditions are used to linked the interaction between these two regions.As we know that almost every engineering problem modeled via PDEs.The analytical solutions of these kind of problems are not easy,so we use numerical approximations.In this study we discuss the two essential properties,namely mass conservation and stability analysis of two types of coupling interface conditions for the oceanatmosphere model.The coupling conditions arise in general circulation models used in climate simulations.The two coupling conditions are the Dirichlet-Neumann and bulk interface conditions.For the stability analysis,we use the Godunov-Ryabenkii theory of normal-mode analysis.The main empha-sis of this work is to study the numerical properties of coupling conditions and an important point is to maintain conservativity of the overall scheme.Furthermore,for the numerical approximation we use two methods,an explicit and implicit couplings.The implicit coupling have further two algorithms,monolithic algorithm and partitioned iterative algorithm.The partitioned iterative approach is complex as compared to the monolithic approach.In addition,the comparison of the numerical results are exhibited through graphical illustration and simulation results are drafted in tabular form to validate our theoretical investigation.The novel characteristics of the findings from this paper can be of great importance in science and ocean engineering.

A sharp-interface immersed smoothed point interpolation method with improved mass conservation for fluid-structure interaction problems

To solve the problem of inaccurate boundary identification and to eliminate the spurious pressure oscillation in the previously developed immersed smoothed point interpolation method(IS-PIM),a new sharp-interface IS-PIM combining mass conservation algorithm,called Sharp-ISPIM-Mass,is proposed in this work.Based on the so called sharp-interface method,the technique of quadratic local velocity reconstruction has been developed by combining with the mass conservation algorithm,which enables the present method improve the accuracy of the velocity field and satisfy the mass conservation condition near the boundary field.So the proposed method would not encounter the problem of spurious mass flux.In addition,a new form of FSI force evaluation considering pressure and viscous force to perform a whole function from the fluid domain to fictitious fluid domain is introduced,which makes the present method obtain more accurate results of FSI force than the original one.Through the numerical studies of a number of benchmark examples,the performance of the Sharp-ISPIM-Mass has been examined and illustrated.

Effects of a Dry-Mass Conserving Dynamical Core on the Simulation of Tropical Cyclones

The accurate forecasting of tropical cyclones(TCs)is a challenging task.The purpose of this study was to investigate the effects of a dry-mass conserving(DMC)hydrostatic global spectral dynamical core on TC simulation.Experiments were conducted with DMC and total(moist)mass conserving(TMC)dynamical cores.The TC forecast performance was first evaluated considering 20 TCs in the West Pacific region observed during the 2020 typhoon season.The impacts of the DMC dynamical core on forecasts of individual TCs were then estimated.The DMC dynamical core improved both the track and intensity forecasts,and the TC intensity forecast improvement was much greater than the TC track forecast improvement.Sensitivity simulations indicated that the DMC dynamical core-simulated TC intensity was stronger regardless of the forecast lead time.In the DMC dynamical core experiments,three-dimensional winds and warm and moist cores were consistently enhanced with the TC intensity.Drier air in the boundary inflow layer was found in the DMC dynamical core experiments at the early simulation times.Water vapor mixing ratio budget analysis indicated that this mainly depended on the simulated vertical velocity.Higher updraft above the boundary layer yielded a drier boundary layer,resulting in surface latent heat flux(SLHF)enhancement,the major energy source of TC intensification.The higher DMC dynamical core-simulated updraft in the inner core caused a higher net surface rain rate,producing higher net internal atmospheric diabatic heating and increasing the TC intensity.These results indicate that the stronger DMC dynamical coresimulated TCs are mainly related to the higher DMC vertical velocity.

CONSERVATION OF MASS FOR A PARTICLEMOVING WITH HIGH VELOCITY

By using the revision of the momentum for a particle moving with high velocity and by investigating the famous Bucherer's experiment of an electron deflecting with high velocity in the electromagnetic fields in 1908, the paper determines that mass of the electron with high velocity is still to observe the law of conservation of mass.

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds transport theorem was presented. Combining these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual localization.

DISCREPANCY OF THE GLOBAL AIR MASS AND WATER BUDGETS AMONG 20 CMIP5 CLIMATE MODELS

The consistency of global atmospheric mass and water budget performance in 20 state-of-the-art ocean-atmosphere Coupled Model Intercomparison Project Phase 5(CMIP5) coupled models has been assessed in a historical experiment. All the models realistically reproduce a climatological annual mean of global air mass(AM) close to the ERA-Interim AM during 1989-2005. Surprisingly, the global AM in half of the models shows nearly no seasonal variation,which does not agree with the seasonal processes of global precipitable water or water vapor, given the mass conservation constraint. To better understand the inconsistencies, we evaluated the seasonal cycles of global AM tendency and water vapor source(evaporation minus precipitation). The results suggest that the inconsistencies result from the poor balance between global AM tendency and water vapor source based on the global AM budget equation. Moreover, the cross-equatorial dry air mass flux, or hemispheric dry mass divergence, is not well represented in any of the 20 CMIP5 models, which show a poorly matched seasonal cycle and notably larger amplitude, compared with the hemispheric tendencies of dry AM in both the Northern Hemisphere and Southern Hemisphere. Pronounced erroneous estimations of tropical precipitation also occur in these models. We speculate that the large inaccuracy of precipitation and possibly evaporation in the tropics is one of the key factors for the inconsistent cross-equatorial mass flux. A reasonable cross-equatorial mass flux in well-balanced hemispheric air mass and moisture budgets remains a challenge for both reanalysis assimilation systems and climate modeling.

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem.The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order.The concentration equation is solved by the method of mixed finite element,where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method.The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation.In actual computations the characteristics adopts a large time step without any loss of accuracy.The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux.In order to derive the optimal 3/2-order error estimate in L^(2) norm,a post-processing technique is included in the approximation to the scalar unknowns.Numerical experiments are illustrated finally to validate theoretical analysis and efficiency.This method can be used to solve such an important problem.

Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem

This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure are derived. A numerical example is presented to verify the theoretical predictions.

MASS CONSERVATION BEHAVIOR OF WAVE EQUATION MODEL FOR SOLVING SHALLOW WATER EQUATIONS

Wave equation model (WEM) first developed by Lynch and Gray [2] is one of accurate and effective numerical methods to resolve shallow water equations. This paper shows the numerical consistency of the second-order wave equation and the first-order continuity equation, analyzes the error between them. This paper also shows that the numerical friction factor τ0 appearing in wave equation is of key importance to the numerical solutions and mass conservation of wave equation model. Numerical calculations of M2 tidal waves in rectangular harbor and a quarter annular harbor are made to demonstrate that it is possible to find a proper numerical friction factor To with which accurate solutions and satisfactory mass conservation can be achieved by wave equation model.

Finite Element Methods for Coupled Stokes and Darcy Problems

We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H（div） conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy equations and a stabilized H（div） finite element method for the Stokes equations. Optimal error estimates for the fluid velocity and pressure are derived. The finite element solutions from the new scheme not only feature a full satisfaction of the continuity equation, which is highly demanded in scientific computing, but also satisfy the mass conservation.

Third-order unconditional positivity-preserving schemes for reactive flows keeping both mass and mole balance

In this paper,the previously proposed second-order process-based modified Patankar Runge-Kutta schemes are extended to the third order of accuracy.Owing to the process-based implicit handling of reactive source terms,the mass conservation,mole balance and energy conservation are kept simultaneously while the positivity for the density and pressure is preserved unconditionally even with stiff reaction networks.It is proved that the first-order truncation terms for the Patankar coefficients must be zero to achieve a prior third order of accuracy for most cases.A twostage Patankar procedure for each Runge-Kutta step is designed to eliminate the first-order truncation terms,accomplish the prior third order of accuracy and maximize the Courant number which the total variational diminishing property requires.With the same approach as the second-order schemes,the third-order ones are applied to Euler equations with chemical reactive source terms.Numerical studies including both 1D and 2D ordinary and partial differential equations are conducted to affirm both the prior order of accuracy and the positivity-preserving property for the density and pressure.

Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin method

In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.

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.

A BLOCK-CENTERED UPWIND APPROXIMATION OF THE SEMICONDUCTOR DEVICE PROBLEM ON A DYNAMICALLY CHANGING MESH

The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initialboundary nonlinear system of four partial differential equations: an elliptic equation for electric potential, two convection-diffusion equations for electron concentration and hole concentration, and a heat conduction equation for temperature. The first equation is solved by the conservative block-centered method. The concentrations and temperature are computed by the block-centered upwind difference method on a changing mesh, where the block-centered method and upwind approximation are used to discretize the diffusion and convection, respectively. The computations on a changing mesh show very well the local special properties nearby the P-N junction. The upwind scheme is applied to approximate the convection, and numerical dispersion and nonphysical oscillation are avoided. The block-centered difference computes concentrations, temperature, and their adjoint vector functions simultaneously.The local conservation of mass, an important rule in the numerical simulation of a semiconductor device, is preserved during the computations. An optimal order convergence is obtained. Numerical examples are provided to show efficiency and application.

CONVERGENCE ANALYSIS OF MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT FOR THREE-DIMENSIONAL CHEMICAL OIL-RECOVERY SEEPAGE COUPLED PROBLEM

The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in 12 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

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.

INVARIANTS-PRESERVING DU FORT-FRANKEL SCHEMES AND THEIR ANALYSES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH WAVE OPERATOR

Du Fort-Frankel finite difference method(FDM)was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953.It is an explicit and unconditionally von Neumann stable scheme.However,there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods(FDMs).In this study,a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D)and two-dimensional(2D)nonlinear Schrödinger equations with wave operator.By using the discrete energy method,it is shown that their solutions possess the discrete energy and mass conservative laws,and conditionally converge to exact solutions with an order of for ofο(T^(2)+h_(x)^(2)+(T/h_(x))^(2))1D problem and an order ofο(T^(2)+h_(x)^(2)+h_(Y)^(2)+(T/h_(X))^(2)+(T/h_(y))^(2))for 2D problem in H1-norm.Here,τdenotes time-step size,while,hx and hy represent spatial meshsizes in x-and y-directions,respectively.Then,by introducing a stabilized term,a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses,but also own much better stability than original schemes.Finally,numerical results demonstrate the theoretical analyses.

Mathematical Model,Numerical Simulation and Convergence Analysis of a Semiconductor Device Problem with Heat and Magnetic Influences

In this paper,the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description,numerical simulation and theoretical analysis.Two important factors,heat and magnetic influences are involved.The mathematical model is formulated by four nonlinear partial differential equations(PDEs),determining four major physical variables.The influences of magnetic fields are supposed to be weak,and the strength is parallel to the z-axis.The elliptic equation is treated by a block-centered method,and the law of conservation is preserved.The computational accuracy is improved one order.Other equations are convection-dominated,thus are approximated by upwind block-centered differences.Upwind difference can eliminate numerical dispersion and nonphysical oscillation.The diffusion is approximated by the block-centered difference,while the convection term is treated by upwind approximation.Furthermore,the unknowns and adjoint functions are computed at the same time.These characters play important roles in numerical computations of conductor device problems.Using the theories of priori analysis such as energy estimates,the principle of duality and mathematical inductions,an optimal estimates result is obtained.Then a composite numerical method is shown for solving this problem.

A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrodinger Equation

Abstract.In this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic form.We use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time discretization.Both mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method.Numerical examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.

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.