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Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type
(2023-12-01)
Sakkaplangkul, Puttha
In this paper, two new effective operator splitting methods (SS-MD and SM-MD) for the Maxwell’s equations for dispersive media in two dimensions transverse electric polarization (the 2D Maxwell–Debye TE model) are presented and analyzed. The splitting schemes consist of two sub-stages in each time step, each of which requires solving a number of 1D discrete sub-problems. The Crank–Nicolson approach is used to solve each sub-problem’s time discretization. Both splitting methods satisfy the energy decay and are unconditionally stable. The convergence result of the SS-MD scheme is shown to be of first order in time and of second order in space based on the energy technique, whereas the SM-MD scheme is of second order in both time and space. We also analyze numerical dispersion analysis to obtain two identities of the discrete numerical dispersion relations of both splitting schemes. Examples and numerical experiments are provided to demonstrate and support our theoretical results.
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A Couple Mathematical Models of the Water Quality Measurement in a Stream using Upwind Implicit Methods
(2021-01-01)
Pochai, N.
;
Phosri, P.
Two numerical simulations are being used to quality of the water in a non-uniform flow stream. The first model is a hydrodynamic model that uses the Crank-Nicolson formula to provide velocity profile and level of water. The second phase is a dispersion model, where the governing function uses advection-dispersion-reaction equations to provide the concentrations of contaminants. The first and second models are described as one-dimensional equations. The first determined flow velocity profile of the hydrodynamic model shall be the input into the dispersion model at each phase. The finite difference methods are proposed to solve the dispersion model a four points explicit upwind schemes, a third order Crank-Nicolson schemes, and the four points implicit methods, which give the approximated pollutant concentrations. Finally, we present a numerical simulation of all schemes, so as to illustrate their applicability to real-world problems. The proposed topic is related and economic to be used in real-world challenges relatively low cost of the program and the straight-forwardness of its implementations. It is also good to achieve suitable places and easier timeframes for different discharge locations.
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Numerical simulation of water-quality model on flooding using revised lax-diffusive and modified siemieniuch-gladwell methods
(2019-12-01)
Subklay, Kanawoot
;
Pochai, Nopparat
In 2011, Thailand has been confronted a largest flooding. The mass of water has been drenched from many main and branch rivers to cover wide areas. The residents who lived in the flooding area have to build a manmade sandbag dike to protect their village. The flooding has been taken for a long time meanwhile the flooding water becomes contaminated. There are some residents in their flooding area want to drain their contaminated water to a nearest area. They have been destroyed their sandbag dike. Consequently, the dispute among residents is occurred. In this research, a mathematical simulation of a water-quality on a long period flooding using a couple of two models is proposed. The first model is the one-dimensional shallow water equations that provide the water elevation and velocity. The second model is a one-dimensional advection-dispersion equation that provides the water pollutant concentrations after the sandbag dike has been destroyed. A revised Lax-diffusive is used to approximate the solution of the first model. Consequently, the numerical solutions of the second model are obtained by using the traditional and modified Siemieniuch-Gladwell schemes.
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Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media
(2019-10-01)
Jiang, Yan
;
Sakkaplangkul, Puttha
;
Bokil, Vrushali A.
;
Cheng, Yingda
;
Li, Fengyan
In this paper, we consider Maxwell's equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell's equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors. Finally, we highlight the limitations of the second order accurate temporal discretizations considered.
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Numerical simulations of a water quality model in a flooding stream due to dam-break problem using implicit and explicit methods
(2017-02-17)
Subklay, Kanawoot
;
Pochai, Nopparat
The heavy monsoon rains have been drenched flooding along the rivers and reservoirs. The residents try to construct the sandbag dike to protect their village. If the residents have encountered flood for long period, the water pollutant must be increase. The villagers want to drain the water to another areas. The residents of another village are not agree to receive the drained-polluted water from them. The simulation of water-quality is required to compromise these problem. The simulation process of water-quality model is require the input as the water flow velocities after the villagers dike has been destroyed but there is lack of field data on the flooding period. In this research, the dam-break model is used to describe unsteady dike failure flow. A couple of mathematical models is used to simulate water-quality in the problem. The first model is the dam-break model that provides the velocity fields and elevation of water. The second model is the dispersion model that provides the pollutant concentration fields. A modified Lax-diffusive method for solving a dam-break model is proposed. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. The explicit and implicit methods are subsequently employed in the dispersion model. This paper proposes a simply remarkable alteration to the Lax-diffusive method, the implicit methods and the explicit methods so as to make it more accurate without any significant loss of computational efficiency. The numerical techniques indicate that this model can be applied for simulation of water-quality in long-term flooding cases are proposed.
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A numerical computation of a non-dimensional form of stream water quality model with hydrodynamic advection-dispersion-reaction equations
(2009-11-01)
Pochai, Nopparat
Mathematical models of water quality assessment problems often arise in environmental science. The modelling often involves numerical methods to solve the equations. In this research, two mathematical models are used to simulate pollution due to sewage effluent in the nonuniform flow of water in a stream with varied current velocity. The first is a hydrodynamic model that provides the velocity field and elevation of the water flow. The second is a dispersion model, where the commonly used governing factor is the one-dimensional advection-dispersion-reaction equation that gives the pollutant concentration fields. In the simulation processes, we used the Crank-Nicolson method system of a hydrodynamic model and the backward time central space scheme for the dispersion model. Finally, we present a numerical simulation that confirms the results of the techniques. © 2009 Elsevier Ltd. All rights reserved.

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