Characteristic Finite Difference Research Articles

The Splitting Characteristic Finite Difference Domain Decomposition Scheme for Solving Time-Fractional MIM Nonlinear Advection–Diffusion Equations

The Splitting Characteristic Finite Difference Domain Decomposition Scheme for Solving Time-Fractional MIM Nonlinear Advection–Diffusion Equations

A Mass-Preserving Characteristic Finite Difference Method for Miscible Displacement Problem

A Mass-Preserving Characteristic Finite Difference Method for Miscible Displacement Problem

Advancing parameter estimation with Characteristic Finite Difference Method (CFDM) for a marine ecosystem model by assimilating satellite observations: Spatial distributions

The ecosystem parameters are critical for precisely determining the marine ecological process and improving the simulations of the marine ecological model. In this study, based on the NPZD (nutrient, phytoplankton, zooplankton and detritus) model, the surface chlorophyll-a observations obtained from Sea-viewing Wide Field-of-view Sensor (SeaWiFS) data were assimilated to estimate spatially ecosystem parameters in the Bohai, Yellow, and East China Seas using an adjoint assimilation method with characteristic finite difference scheme. The experiments of the moving Gaussian hump indicated that the characteristic finite difference method (CFDM) can get rid of the limit of stability and permit using large time steps, which reduces long computation durations and large memory requirements. The model performance was significantly improved after data assimilation with CFDM using a large time step of 6 hours. Moreover, the distributions of parameters of the NPZD model in winter in the Bohai Sea, the Yellow Sea, and the East China Sea were simulated by our method. Overall, the developed method can efficiently optimize the ecosystem parameters and the results can be beneficial for determining reasonable parameters of the marine ecological model.

Advancing Three-Dimensional Coupled Water Quality Model of Marine Ranches: Model Development, Global Sensitivity Analysis, and Optimization Based on Observation System

Marine ranching is a stock enhancement project that has been an important part of aquaculture in China. Due to the lack of scientific management, disasters have occurred, resulting in millions of economic losses. Based on the observation system of marine ranches, a three-dimensional water quality model of marine ranches was developed to study the spatio-temporal variation of key ecological indicators, including the concentrations of chlorophyll-a, zooplankton, detritus, nutrients, and dissolved oxygen (DO). The model is coupled offline with the residual current, temperature, and salinity simulated by a regional oceanic modeling system (ROMS). The conservative characteristic finite difference (C-CFD) scheme is introduced to solve the equations, which guarantees model stability and mass conservation and allows for a larger time step compared to traditional difference schemes. In state-of-the-art water quality models, the biogeochemical processes are parameterized. Due to the complexity of the water quality model, a combination of global sensitivity analysis (GSA) and the adjoint method is introduced as the methodology to optimize the model parameters. Morris’ sampling method is implemented as the GSA method to find out the key factors of the water quality model. The optimization of sensitive parameters with the adjoint method significantly improves the model precision, while the other parameters can be set as empirical values. The results indicate that the combination of GSA and the adjoint method is efficient in the parameter optimization of the water quality model. The model is applicable in marine ranches.

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations

Mass conservative characteristic finite difference method for convection–diffusion equations

In the paper, the space second-order conservative characteristic finite difference method for solving convection–diffusion equations is developed and analysed. Applying the directional derivative and mass correction technique, the convection–diffusion equations are changed into the parabolic equations, where the convection term and unsteady term are considered as one term. The splitting implicit solution-flux coupled scheme computes the solutions and fluxes on the staggered meshes. The mass conservation, stability and convergence of our proposed characteristic difference method are analysed theoretically. Numerical experiments are presented to illustrate mass conservation and convergence.

An efficient characteristic finite difference S-DDM scheme for convection–diffusion equations

In this paper, an efficient splitting domain decomposition method scheme for solving time-dependent convection–diffusion reaction equations is analyzed. A three-step method along each direction is used to solve the solution over each block-divided sub-domain at every time interval. The new solutions are firstly solved by the quadratic interpolation. Then, the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the new solutions. Lastly, the solutions and fluxes in the interiors of sub-domains are computed by the splitting implicit characteristic difference method. By some auxiliary lemmas and the defined intermediate exact solution, the stability and error estimate are given in discrete L2-norm. We further prove that our scheme is of second-order convergence in space and of first-order convergence in time. Numerical experiments are presented to validate theoretical result.

Balance and Characteristic Finite Difference Schemes for Equations of the Parabolic Type

A new numerical algorithm for solving parabolic equations by balance & characteristic difference schemes is proposed. The balance and characteristic schemes combine the advantages of conservative and characteristic schemes. The main advantage of the new algorithm is that it is explicit and is implemented on the most compact computational template in one space-time computational cell. It scales perfectly and is extremely economical in software systems that implement the CABARET scheme because it uses the existing data structure there.

A characteristic ADI finite difference method for spatial-fractional convection-dominated diffusion equation

Based on characteristic method and shifted Grünwald fractional difference method, a characteristic finite difference method is proposed for solving the one/two/three-dimension spatial-fractional convection-dominated diffusion equation. The resulting schemes are first-order accuracy in time and second-order accuracy in space. For high-dimensional problems, alternating direction implicit (ADI) schemes are further proposed. The stability and convergence properties of these schemes are discussed. Numerical experiments are carried out to support the theoretical analysis, and some comparisons with the implicit upwind finite difference scheme are presented to show the effectiveness of the proposed method.

A Study of the Transport of Marine Pollutants Using Adjoint Method of Data Assimilation with Method of Characteristics

An adjoint method of data assimilation with the characteristic finite difference (CFD) scheme is applied to marine pollutant transport problems and the temporal and spatial distribution of marine pollutants are simulated. Numerical tests of two-dimensional problems of pollutant transport with two different schemes indicate that the error of CFD is smaller than that of central difference scheme (CDS). Then the inversion experiments of the initial field and the source and sink terms of pollutants are carried out. Applying CFD in the adjoint method of data assimilation cannot only reduce simulation error to get a good inversion but can also enable larger time step size to decrease computation time and improve the calculation efficiency.

Finite difference scheme for multi-term variable-order fractional diffusion equation

In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order alpha(x,t) in(0,1) and the Riesz variable-order space fractional derivatives of order beta(x,t) in (0,1), gamma(x,t)in(1,2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.

The Time Second Order Mass Conservative Characteristic FDM for Advection–Diffusion Equations in High Dimensions

We propose in this paper a time second order mass conservative algorithm for solving advection–diffusion equations. A conservative interpolation and a continuous discrete flux are coupled to the characteristic finite difference method, which enables using large time step size in computation. The advection–diffusion equations are first transformed to the characteristic form, for which the integration over the irregular tracking cells at previous time level is proposed to be computed using conservative interpolation. In order to get second order in time solution, we treat the diffusion terms by taking the average along the characteristics and use high order accurate discrete flux that are continuous at tracking cell boundaries to obtain mass conservative solution. We demonstrate the second order temporal and spatial accuracy, as well as mass conservation property by comparing results with exact solutions. Comparisons with standard characteristic finite difference methods show the excellent performance of our method that it can get much more stable and accurate solutions and avoid non-physical numerical oscillation.

Application of Adjoint Data Assimilation Method to Atmospheric Aerosol Transport Problems

We propose combining the adjoint assimilation method with characteristic finite difference scheme (CFD) to solve the aerosol transport problems, which can predict the distribution of atmospheric aerosols efficiently by using large time steps. Firstly, the characteristic finite difference scheme (CFD) is tested to compute the Gaussian hump using large time step sizes and is compared with the first-order upwind scheme (US1) using small time steps; the US1 method gets E2 error of 0.2887 using Δt=1/450, while CFD method gets a much smaller E2 of 0.2280 using a much larger time step Δt=1/45. Then, the initial distribution of PM2.5 concentration is inverted by the adjoint assimilation method with CFD and US1. The adjoint assimilation method with CFD gets better accuracy than adjoint assimilation method with US1 while adjoint assimilation method with CFD costs much less computational time. Further, a real case of PM2.5 concentration distribution in China during the APEC 2014 is simulated by using adjoint assimilation method with CFD. The simulation results are in good agreement with the observed values. The adjoint assimilation method with CFD can solve large scale aerosol transport problem efficiently.

Modelling multi-component aerosol transport problems by the efficient splitting characteristic method

In this paper, a splitting characteristic method is developed for solving general multi-component aerosol transports in atmosphere, which can efficiently compute the aerosol transports by using large time step sizes. The proposed characteristic finite difference method (C-FDM) can solve the multi-component aerosol distributions in high dimensional domains over large ranges of concentrations and for different aerosol types. The C-FDM is first tested to compute the moving of a Gaussian concentration hump. Comparing with the Runge-Kutta method (RKM), our C-FDM can use very large time step sizes. Using Δt = 0.1, the accuracy of our C-FDM is 10−4, but the RKM only gets the accuracy of 10−2 using a small Δt = 0.01 and the accuracy of 10−3 even using a much smaller Δt = 0.002. A simulation of sulfate transport in a varying wind field is then carried out by the splitting C-FDM, where the sulfate pollution is numerically showed expanding along the wind direction and the effects of the different time step sizes and different wind speeds are analyzed. Further, a realistic multi-component aerosol transport over an area in northeastern United States is studied. Concentrations of PM2.5 sulfate, ammonium, nitrate are high in the urban area, and low in the marine area, while sea salts of sodium and chloride mainly exist in the marine area. The normalized mean bias and the normalized mean error of the predicted PM2.5 concentrations are −6.5% and 24.1% compared to the observed data measured at monitor stations. The time series of numerical aerosol concentration distribution show that the strong winds can move the aerosol concentration peaks horizontally for a long distance, such as from the urban area to the rural area and from the marine area to the urban and rural area. Moreover, we also show the numerical time duration patterns of the aerosol concentration distributions due to the affections of the turbulence and the deposition removal. The developed splitting C-FDM algorithm can be applied to model spatial multi-component aerosol transport problems in large domains in atmosphere.

Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation

ABSTRACTIn this article, two characteristic block-centred finite difference schemes are introduced and analysed to solve the nonlinear convection-dominated diffusion equation. One scheme is a linear difference approximation which shows that the discrete and errors are , while the other is a two-grid scheme which demonstrates that the discrete and errors are , where h corresponds to a finer grid and H corresponds to a coarser grid. Error estimates with both schemes are established on a non-uniform rectangular grid. Finally, numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis and validate the efficiency of the two-grid method.

The conservative characteristic FD methods for atmospheric aerosol transport problems

In the paper, we develop the new conservative characteristic finite difference methods (C-CFD) for the atmospheric aerosol transport problems. We propose the time second-order and spatial high-order conservative characteristic finite difference methods for the aerosol vertical advection-diffusion process and the two-dimensional conservative characteristic finite difference methods for aerosol horizontal transport process in the second-order splitting algorithm. Based on the characteristic form of advection-diffusion equations tracking back along the characteristic curve, we treat the integrals over the tracking cells at the previous time level by the conservative interpolations and propose to treat the diffusion terms by the average along the characteristics, where the high-order discrete fluxes are obtained by approximating the cumulative mass function and are continuous at the tracking points. The important feature is that the proposed C-CFD schemes preserve mass and have second-order accuracy in time and high-order accuracy in space. Numerical tests are taken to show the accuracy in time and space and mass conservation of our C-CFD schemes, compared with the standard CFD method. A real case of air quality modelling during the 2008 Beijing Olympics and a severe haze in North China are further simulated and analyzed by using our C-CFD algorithm. Simulated results are in good agreement with observations. The developed C-CFD algorithm can be used for efficiently solving large scale atmospheric aerosol transport problems.

Theory and application of fractional step characteristic finite difference method in numerical simulation of second order enhanced oil production

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

Theory and Application of Characteristic Finite Difference Fractional Step Method of Capillary Force Enhanced Oil Production

A kind of second-order implicit characteristic fractional steps finite difference method is presented in this paper for the numerical simulation coupled system of enhanced (chemical) oil production on consideration capillary force in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in $l^2$ norm is derived. This method has been applied successfully the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

A block-centered characteristic finite difference method for convection-dominated diffusion equation

In this paper, a block-centered characteristic finite difference method is proposed for solving the convection-dominated diffusion equation on nonuniform grids. The resulting scheme is first-order accurate in time and second-order accurate in space. The stability analysis and error estimate are deduced rigorously. And the numerical solutions of unknown variable along with its first derivatives are also obtained. Finally, numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the proposed method. Moreover, we show that numerical results using non-uniform grids are significantly more accurate than those using uniform grids for high gradient problems.

Characteristic fractional step finite difference method for nonlinear section coupled system

For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some techniques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in l2 norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.