| In recent decades, ground water resources have been seriously threat-ened by the leaching contaminants from industrial waste, agricultural fer-tilizers and domestic sewage, etc. Thus, there are some research about the multicomponent contamination flows in groundwater (see [11,16,31,35, 52,74,76,79,80,84,102,103], etc). In groundwater, the each compo-nent is transported by advection, molecular diffusion and mechanical disper-sion. Besides, each component is influenced by geochemical reactions between the multicomponents. In general, the reaction between the components can be described as kinetically controlled dissolution-precipitation reaction (see [12,19,38,39,44,49,56,63,64,70.78], etc) or the geochemical equilibrium reactions as hydrolysis aqueous complexation, oxidation-reduction, ion ex-change, surface complexation, and gas dissolution-exsolution reactions (see [14,45,51,65,96,97], etc). Modeling multicomponent transport in ground-water systems (see as [18,64,71],etc) requires to solve the coupled and non-linear time-dependent partial differential equations (the parabolic equations for the water head and the advection-diffusion equations for multicomponent concentration). Numerical methods (FEM, FDM, etc) are important tools to simulate multicomponent transport processes (see [9,10,30,40.48,50,77, 89,98,100,102,103], etc). Due to coupled and nonlinear of partial differen-tial equations, large scale simulations and the long computational time, it is important to develop efficient mass-preserving domain decomposition meth-ods for solving the coupling of water head and multicomponent concentration equations in groundwater.Domain decomposition methods (see [1,2,3,4,15,27,32,72,85,91], etc) allow the reduction of the sizes of problems by dividing the large domain into smaller ones on which the PDEs can be solved by multiple computers in parallel, including the overlapping domain decomposition methods (see [33,60,61,85,94], etc) and non-overlapping domain decomposition methods (see [5,20,21,24,25,26,32,53,55,57,82,83,101,108,110], etc). Since non-overlapping methods have low computation and communication costs at each time step, the non-iterative explicit-implicit schemes on non-overlapping domain decompositions have been developed for solving large scale problems of parabolic types. Papers [20,53] proposed a mixed/hybrid schemes, where the interface solutions were first computed by the explicit schemes on in-terfaces, then the interior solutions were solved by the implicit schemes in sub-domains. In order to relax the stability requirements, paper [26] fur-ther proposed the explicit-implicit domain decomposition (EIDD) methods for solving parabolic problems, where a multi-step explicit scheme or a high-order explicit scheme was used on the interfaces. By introducing the implicit correction step on the interfaces, papers [83,110] proposed the uncondition-al stability EIDD methods for solving the parabolic equations. Paper [83] studied a class of corrected explicit-implicit domain decomposition (CEIDD) method, where the zigzag-shaped was analyzed theoretically for solving two dimensional heat equations. Paper [82] proposed a time three level scheme for solving the parabolic equations, where the inner boundary solutions were firstly computed by the values of previous two time levels at the interface points, then the values in the sub-domain were computed by the fully implic-it schemes, and the inner boundary solutions were updated by the implicit schemes. Recently, paper [57] proposed an efficient explicit-implicit splitting-domain decomposition method (S-DDM) for solving the parabolic equations over multi-block sub-domains by combining the non-overlapping domain de-composition and the splitting technique, where an efficient local multilevel scheme was used for computing the values on the interfaces of sub-domains, and the interior values were computed by one dimensional splitting implicit scheme. With respect to the domain decomposition methods for convection-diffusion equations, paper [108] proposed a time three level non-overlapping domain decomposition method for solving time-dependent variable coeffi-cient convection-diffusion equations in two dimensions. Paper [55] proposed the non-iterative type of domain decomposition methods for solving the convection-diffusion equations over stripe-divided sub-domains, where the solutions on interior boundaries were computed by the explicit scheme, and the solutions in sub-domains were computed by the implicit modified upwind scheme in parallel. By combining the non-overlapping domain decomposi-tion, the splitting technique and one order upwind scheme, paper [25] pro-posed an efficient explicit-implicit splitting-domain decomposition method (S-DDM) for solving the convection-diffusion equations over multi-block sub-domains in high dimensions. However, these previous explicit-implicit do-main decomposition methods [20,24,25,26,53,55,57,82,83,108,110] broke the physical law of mass conservation over multiple sub-domains.Preserving mass of numerical domain decomposition schemes is impor-tant and required for solving the real applications in parallel computing, es-pecially important for long time simulations and large scale problems. Paper [21] presented an explicit-implicit conservative domain decomposition proce-dure for parabolic equations in one dimension and over stripe-divided sub-domains along one variable in two dimensions, where fluxes at sub-domain interfaces were calculated by a average operator from the solutions at the previous time level. The paper proved the scheme to be conditionally stable and obtained the error estimate (not optimal) over the stripe-divided sub-domains in two dimensions. Paper [109] proposed the mass-conserved domain decomposition method for one dimensional constant coefficient parabolic e-quations. It satisfy mass conservation and is unconditional stability, but it can not be extended to solve two dimensional parabolic problems over multi-block sub-domains. The schemes [21,109] are only solving the parabolic equations in one dimension and over stripe-divided sub-domains in two di-mensions. So, it is a challenging and practical significance task to study the mass-preserving domain decomposition methods on multi-block sub-domains in parallel computing.The dissertation is divided into four chapters. The outline is as follows:In Chapter 1, we propose the mass-preserving splitting-domain decom-position method for solving two-dimensional parabolic equations with Neu-mann boundary conditions. Parabolic equations are used to model ground- water water-head, oil reservoir pressure and heat temperature, etc. Due to many large scale applications and long computational time, it is required to develop the mass-preserving domain decomposition methods for solving the parabolic equations. On the non-overlapping block-divided domain de-composition, by combining the splitting technique and the coupling of the solutions and its fluxes on staggered meshes, we propose and analyze the mass-preserving splitting-domain decomposition method (S-DDM) for solv-ing two-dimensional parabolic equations. In the method, the global domain is divided into multi-block non-overlapping sub-domains. At each time step level, we take two stages to compute the solutions and fluxes on sub-domains. Firstly, the intermediate interface fluxes of x-direction are computed by a semi-implicit (explicit) flux scheme. The intermediate solutions and fluxes in the interiors of sub-domains are computed by the x-directional splitting implicit solution-flux coupled scheme. Further, the intermediate fluxes on the interfaces of sub-domains are recomputed explicitly by the obtained so-lutions in the interiors of sub-domains. Secondly, the interface fluxes of y-diroction are computed by a semi-implicit (explicit) flux scheme. The solu-tions and fluxes are computed by the y-directional splitting implicit solution-flux coupled scheme in the interiors of sub-domains. Then, the final fluxes on the interfaces of sub-domains are corrected explicitly. The proposed mass-preserving S-DDM scheme for parabolic equations is not only in the ad-vantages as non-overlapping block-divided domain decomposition, operator splitting and unconditional stability, what’s more, it is mass conservative over multiple sub-domains. Analyzing the stability and convergence of the scheme over block-divided sub-domains is challenging due to the combination of the splitting technique and the domain decomposition. By deriving out some auxiliary lemmas of estimating numerical fluxes in the interface areas, introducing intermediate variable values of solutions and applying the matrix theory for analyzing truncation errors, we prove the scheme unconditionally stable and mass conservative, and obtain the error estimate in L2-norm over multi-block sub-domains. Numerical experiments confirm theoretical results.In Chapter 2, we propose the mass-preserving domain decomposition method for solving the convection-diffusion equations. Convection-diffusion equations are widely used to describe heat transfer, components mass trans-port and aerodynamics movement, etc. In many such applications, the con-vection terms essentially dominate diffusion terms, which leads to nearly hyperbolic set of governing partial differential equations. It is well docu-mented that standard finite difference methods are usually invalid for the numerical oscillation. Much explicit/implicit domain decomposition methods work have been done on the study of the convection-diffusion equations (see [25.55,108]). However, the above domain decomposition methods break the mass conservation. To date, there is no research on mass conservative domain decomposition methods for the convection-diffusion problems. By combining modified upwind technique and the coupling of the solutions and its fluxes on staggered meshes, we first propose a new mass-preserving modified upwind splitting-domain decomposition method (S-DDM) for solving the convection-diffusion equations over multiple non-overlapping block-divided sub-domains. On each block-divided sub-domain, we take two half time steps to compute the final solutions and fluxes. In the first half time step, the intermediate x-direction interface fluxes are first computed by the modified semi-implicit (explicit) flux scheme on the interfaces of sub-domain. The solutions and x-direction fluxes in the interiors of sub-domain are then computed by the x-directional splitting modified upwind implicit solution and flux coupled scheme. Then the intermediate x-direction interface fluxes are corrected ex-plicitly. At the second half time step, the y-direction interface fluxes are eval-uated by the modified semi-implicit (explicit) flux scheme on the interfaces. The solutions and y-direction fluxes are solved by the y-directional splitting modified upwind implicit solution and flux coupled scheme in the interiors of sub-domain. The y-direction interface fluxes are updated by the obtained solutions. We prove theoretically that our scheme preserves mass and is un-conditionally stable in discrete L2-norm over multi-block sub-domains. We prove the convergence and obtain the error estimate. Numerical experiments show mass conservation, convergence and parallelism of our scheme. Our proposed modified upwind S-DDM scheme for solving convection-diffusion equations is not only in the advantages as non-overlapping block-divided do-main decomposition, operator splitting, second-order in space and uncondi-tional stability, but also successfully overcomes the serious non-conservation problems in previous work [25,55,108].In Chapter 3, we analyze a kind of mass-conserved domain decomposi-tion method for solving the parabolic equations with the. diffusion coefficients which dependent on the time and space. By combining the weighted aver-age finite difference scheme on the interface and the splitting technique in x-direction and y-direction, we propose and analyze a kind of mass-conserved S-DDM scheme for solving two-dimensional variable coefficient parabolic e-quations with reaction terms on non-overlapping block-divided domain de-compositions. A fraction step operator splitting method is applied to com-pute the interior solutions over block-divided sub-domains at each time step level. While the.r-direction interface fluxes are computed explicitly by the local multi-point weighted average schemes, and the interior solutions and x-direction fluxes of sub-domains at each time step are computed by the x-direction splitting implicit scheme. Similarly, we can compute the final solutions and y-direction fluxes along y-direction. We prove the proposed weighted average finite difference on the interface scheme to satisfy the glob-al mass conservation over the whole domain. We prove the scheme is stable under a weak condition. We analyze the convergence and prove the optimal error estimate O(△t+hx2+hy2+Hx5/2+Hy5/5/2) in discrete L2-norm. Numer-ical experiments are performed to illustrate convergence, mass conservation, stability, and parallelism. Our developed mass-conserved S-DDM scheme successfully overcomes the limitation of stripe-divided sub-domains in two dimensions in previous work [21,109].In Chapter 4, we study the mass-preserving domain decomposition method with applications in multicomponent contamination flows in groundwater. The mathematic models, which are coupled and nonlinear time-dependent partial differential equations (the parabolic equations for the water head and the advection-diffusion equations for multieomponent concentration), are used to simulate the multieomponent contaminants transport in ground-water. Due to the complexity of the geographic structure, large scale simu-lations and the long computational time, it is important to develop efficient mass-preserving domain decomposition methods for solving the multieompo-nent contamination flows in porous media. Combining the non-overlapping domain decomposition, the splitting technique and the coupling of the so-lutions and its fluxes on staggered meshes, we propose the mass-preserving S-DDM iterative algorithms for solving the water head and multieomponent concentration equations on block-divided domain decomposition in two step-s at each time interval in porous media, respectively. Firstly, the interface fluxes of water head are first computed by the semi-implicit (explicit) fluxes, while the solutions and fluxes of water head in the interiors of sub-domains are computed by one dimensional splitting implicit solution and flux cou-pled scheme. Secondly, using the obtained water head and the linearizing concentration technique, we solve the re-direction and 2-direction Dracy’s ve-locity, respectively. Thirdly, by applying the modified upwind technique and defining the new conserved fluxes method, the mass-preserving modified up-wind S-DDM scheme is proposed to solve the multieomponent concentration equations. Our iterative approach not only keeps the excellent advantages of the non-overlapping block-divided domain decomposition and the opera-tor splitting technique, but also preserves the important physical law of mass conservation. The numerical experiments are presented to illustrate the mass conservation, convergence order, stability and parallelism. We simulate the multieomponent transport and precipitation reaction between contaminants Ca2+, CO\-’, Cl-and Na+in porous media. When the contaminant HCl flows into the porous media which includes CaCO3 and MgCOz, there will produce dissolution reaction and give the new contaminants Ca2+, Mg2+ and H2CO3. Numerical results show that our proposed mass-preserving S-DDM scheme can simulate well multieomponent contamination flows in porous me- dia. |
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