6621687
Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream
Two mathematical models are used to simulate water quality in a nonuniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the elevation of water. The second model is the dispersion model that provides the pollutant concentration field. Both models are formulated in one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. A modified MacCormack method is subsequently employed in the second model. This paper proposes a simply remarkable alteration to the MacCormack method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed modified MacCormack scheme does improve the prediction accuracy compared to that of the traditional MacCormack method.
122979328
A Mathematical Model of Water Pollution Control Using the Finite Element Method
assessment of the chemical oxygen demand(COD) concentration in the river is considered. This model requires the calculation of the substance dispersion given the water velocity in the channel. A finite element model is used to compute the concentration of the pollutant for variable inputs. These are then subjected to the optimal control of the water treatment plants to achieve minimum cost. A numerical example given.
17901643
Role of spatial distribution of porous medium surface charge heterogeneity in colloid transport
Abstract The role of spatial distribution of porous medium patchwise chemical (charge) heterogeneity in colloid transport in packed bed columns is investigated. Colloid transport experiments with carboxyl latex particles flowing through columns packed with chemically heterogeneous sand grains were carried out. Patchwise chemical heterogeneity was introduced to the granular porous medium by modifying the surface chemistry of a fraction of the quartz sand grains via reaction with aminosilane. Colloid transport experiments at various degrees of patchwise charge heterogeneity and several spatial distributions of heterogeneity were conducted at different flow rates and background electrolyte concentrations. Colloid deposition rate coefficients were determined from analysis of particle breakthrough curves as a response to short-pulse colloid injections to the column inlet. Experimental colloid deposition rate coefficients compared well with theoretical predictions based on a colloid transport model that incorporates patchwise chemical heterogeneity. The results revealed the particle deposition rate and transport behavior to be independent of the spatial distribution of porous medium chemical heterogeneity. It is the mean value of chemical heterogeneity rather than its distribution that governs the colloid transport behavior in packed columns.
26039071
Simple, accurate, and efficient revisions to MacCormack and Saulyev schemes: High Peclet numbers
Abstract Stream water quality modeling often involves numerical methods to solve the dynamic one-dimensional advection–dispersion–reaction equations (ADRE). There are numerous explicit and implicit finite difference schemes for solving these problems, and two commonly used schemes are the MacCormack and Saulyev schemes. This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Peclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank–Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes were tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances were considered with respect to stability, accuracy, and computational efficiency. Results indicated that both the modified Saulyev and the MacCormack schemes, which are named the Saulyevc and MacCormackc schemes respectively, greatly improved the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyevc scheme. The MacCormackc scheme demonstrated increased time consumption but was still much faster than implicit schemes.
129307445
DILUTION AND DECAY OF AQUATIC HERBICIDES IN FLOWING CHANNELS
Abstract The behaviour of herbicides injected into flowing irrigation channels for aquatic weed control is analysed. Solutions to the one-dimensional convective-diffusion equation are given for instantaneous and step-function injections of a decaying tracer. The basic solutions are used to derive methods for determining the degradation rate from field experiments, and the total exposure to the herbicide as the water flows over the weeds. It is shown that some significant aspects of herbicide behaviour are unaffected by dispersion, and depend only on the degradation rate constant and the travel time of the material. Observations from trials in irrigation channels in the U.S.A. and Australia with the herbicide acrolein are analysed. The herbicide application rates and computed exposures vary greatly. Degradation rate constants are more nearly equal.
8511393
Numerical schemes for one-dimensional parabolic equations with nonstandard initial condition
Parabolic partial differential equations with nonstandard initial condition, feature in the mathematical modelling of many phenomena. While a significant body of knowledge about the theory and numerical methods for parabolic partial differential equations with classical initial condition has been accumulated, not much has been extended to parabolic partial differential equations with nonstandard initial condition. In this paper finite difference approximations to the solution of one-dimensional parabolic equation with nonstandard initial condition are studied. Several finite difference schemes are presented for solving a parabolic partial differential equation with nonlocal time weighting initial condition. These schemes are based on the forward time centred space explicit formula, the backward time centred space implicit technique, the Crank-Nicolson implicit formula, the Crandall's implicit scheme and the Saulyev's (2,2) explicit methods. The results of a numerical experiment are presented.
97229205
Improving the numerical modeling of river water quality by using high order difference schemes
Abstract A one-dimensional numerical model is presented for the description of water quality in rivers. The model is an improvement over existing models, since it involves two differential schemes of third-order truncation error (QUICK and QUICKEST), which avoid the stability problems of central difference, while remaining relatively free of the inaccuracies of numerical dispersion associated with the upwind difference scheme. The model is successfully applied to four simple, steady-state and transient water quality problems. Results show that the commonly used upwind scheme performs very poorly. The central difference scheme performs satisfactorily in steady-state implicit calculations, but not in transient problems, where it shows numerical dispersion and significant oscillations. Calculations with QUICK and QUICKEST are stable and accurate involving the minimum error and minimum level of numerical dispersion in both steady-state and transient problems.
62641143
Finite element simulation of dispersion in the Bay of Santander
Abstract Two mathematical models are used to simulate pollution in the Bay of Santander. The first is the hydrodynamic model that provides the velocity field and height of the water. The second gives the pollutant concentration field as a resultant. Both models are formulated in two-dimensional equations. Linear triangular finite elements are used in the Galerkin procedure for spatial discretization. A finite difference scheme is used for the time integration. At each time step the calculated results of the first model are input to the second model as field data. The efficiency and accuracy of the models are tested by their application to a simple illustrative example. Finally a case study in simulation of pollution evolution in the Bay of Santander is presented.
119821167
A numerical computation of the non-dimensional form of a non-linear hydrodynamic model in a uniform reservoir
Abstract A mathematical model is used to simulate the water current and the elevation in a uniform reservoir. A non-linear hydrodynamic model that provides the velocity field and elevation of the water flow is considered. In the simulating process, the Lax–Wendroff technique is used to approximate the solutions. The numerical solution can be the input data for a water-quality model that is applicable for the optimal control of water treatment in the system to achieve minimum cost.
129775063
A water quality computation in the uniform channel
Abstract In this research, two mathematical models are used to simulate pollution due to sewage effluent in the uniform channel with varied current velocity. The first is the hydrodynamic model that provides the velocity field and elevation of the water flow. The second is the dispersion model that gives the pollutant concentration field. In the simulating processes, we used the Crank-Nicolson method to system of hydrodynamic model and the backward-implicit and central different technique to the dispersion model. The accuracy of the models are tested by the illustrative example.
121274134
A numerical computation of a non-dimensional form of stream water quality model with hydrodynamic advection-dispersion-reaction equations
Abstract 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.
128658085
Surface Water-Quality Modeling
Since the 1920's, scientists and engineers around the globe have been using mathematical models to simulate the transport and fate of pollutants in natural waters. Today, and in the foreseeable future, more of these applications are being generated in an effort to develop economical solutions to water-quality problems.The primary audience for this book is first-year graduate students, including both MA and Ph.D. students. The book, however, could be used as a basis for a senior undergraduate course. The text is divided into seven major parts. The first two cover Modeling Fundamentals, (including material on mathematics, numerical methods, kinetics, diffusion, etc). The remaining parts deal with major water-quality modeling problems such as dissolved oxygen, eutrophication, and toxics. The text is written in lecture format, ideal for case study and teaching purposes. The book stresses theory and application. This edition has a strong computer orientation with a Visual Basic computer program available on the Internet.
121180749
A Numerical Treatment of Nondimensional Form of Water Quality Model in a Nonuniform Flow Stream Using Saulyev Scheme
The stream water quality model of water quality assessment problems often involves numerical methods to solve the equations. The governing equation of the uniform flow model is one-dimensional advection-dispersion-reaction equations (ADREs). In this paper, a better finite difference scheme for solving ADRE is focused, and the effect of nonuniform water flows in a stream is considered. Two mathematical models are used to simulate pollution due to sewage effluent. The first is a hydrodynamic model that provides the velocity field and elevation of the water flow. The second is a advection-dispersion-reaction model that gives the pollutant concentration fields after input of the velocity data from the hydrodynamic model. For numerical techniques, we used the Crank-Nicolson method for system of a hydrodynamic model and the explicit schemes to the dispersion model. The revised explicit schemes are modified from two computation techniques of uniform flow stream problems: forward time central space (FTCS) and Saulyev schemes for dispersion model. A comparison of both schemes regarding stability aspect is provided so as to illustrate their applicability to the real-world problem.
45750853
A third-order upwind scheme for the advection-diffusion equation using spreadsheets
In this study, numerical solution of advection-diffusion equation with third-order upwind scheme by using spreadsheet simulation (ADE-TUSS) is carried out. This is a user-friendly and a flexible solution algorithm for the numerical solution of the one dimensional advection-diffusion equation (ADE). The ADE-TUSS algorithm is based on the description of ADE by using the third-order upwind scheme (TU) for advection term and second-order central finite representation. For the solution of the governing equations, spreadsheet simulation (SS) technique is used instead of conventional solution techniques. The solution of ADE can be obtained for explicit, implicit and Crank-Nicolson schemes by only changing the values of temporal weighted parameter in the ADE-TUSS algorithm. It is clear in the literature that numerical diffusion causes great deviations in the model results when the first- or second-order upwind schemes are used. In order to decrease the numerical diffusion and to obtain oscillation free results, an artificial diffusion term is usually defined or sizes of the time step and/or grid sizes are set small values. Reduce of the grid sizes and/or time step increases the computational time and generally requires writing a fairly complex code when high order schemes are used. However, numerical solution of ADE by taking into account TU scheme has been carried out by using the iterative spreadsheet solution technique in the proposed solution algorithm. In order to simulate transient solution, a simple macro that carries out time cycle is defined with the help of VBA feature of spreadsheets. One of the most important advantages of ADE-TUSS algorithm is that it does not require the matrix algebra at each time step of the transient solutions. In order to test the ADE-TUSS model, two examples having numerical and analytical solutions are solved. Results showed that use of the high-order schemes in the spreadsheet simulation is very applicable for the numerical solution of ADE. Moreover, numerical diffusion problem is drastically prevented by using the ADE-TUSS model.
115149106
A New Flux Splitting Scheme
A new flux splitting scheme is proposed. The scheme is remarkably simple and yet its accuracy rivals, and in some cases surpasses, that of Roe's solver in the Euler and Navier-Stokes solutions carried out in this study, The scheme is robust and converges as fast as the Roe splitting. We propose an appropriately defined cell-face advection Mach number using values from the two straddling cells via associated characteristic speeds. This interface Mach number is then used to determine the upwind extrapolation for the convective quantities. Accordingly, the name of the scheme is coined as the advection upstream splitting method (AUSM). We also introduce a new pressure splitting which is shown to behave successfully, yielding much smoother results than other existing pressure splittings. Of particular interest is the supersonic blunt body problem in which the Roe scheme gives anomalous solutions. The AUSM produces correct solutions without difficulty for a wide range of flow conditions as well as grids.
1670025
Godunov Mixed Methods on Triangular Grids for Advection–Dispersion Equations
A time-splitting approach for advection–dispersion equations is considered. The dispersive and advective fluxes are split into two separate partial differential equations (PDEs), one containing the dispersive term and the other one the advective term. On triangular elements a triangle-based high resolution Finite Volume (FV) scheme for advection is combined with a Mixed Hybrid Finite Element (MHFE) technique to solve dispersion. This approach introduces an error proportional to the time step and the overall scheme is only first order accurate if special care is not taken in the definition of the numerical flux approximation for advection. By incorporating the diffusive effects into the definition of this numerical flux, near second order accuracy (up to a log h factor) can be proved theoretically and validated by numerical experiments in both one- and two-dimensional cases.
123625057
Godunov-mixed methods for advection-diffusion equations in multidimensions
Time-split methods for multidimensional advection-diffusion equations are considered. In these methods, advection is approximated by a Godunov-type procedure, and diffusion is approximated by a low...
203224
Flux-limiting solution techniques for simulation of reaction–diffusion–convection system
The aim of this work is to analyze the use of a number of flux-limiters to simulate numerically the behavior of a homogeneous tubular reactor which exhibits steep moving fronts. The strength and limitations of five different flux-limiters are examined for different test cases. All flux-limiters are found successful in capturing the steep concentration profiles. The simulations show that the minmod and MUSCL flux-limiter are more appropriate for the convection–diffusion case. On the other hand, the superbee limiter shows the best performance in capturing discontinuities in the convection–diffusion–reaction case. 2005 Elsevier B.V. All rights reserved.
120505408
Total Variation Diminishing Implicit Runge‐Kutta Methods for Dissipative Advection‐Diffusion Problems in Astrophysics
Brandenburg University of Technology Cottbus, Department of Aerodynamics and Fluid Mechanics,D-03046 Cottbus, GermanyWe investigate the properties of dissipative full discretizations for the equations of motion associated with models of ﬂowand radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with speciallyadapted total-variation-diminishing (TVD) or strongly stable Runge–Kutta time discretizations with adaptive step-size controlthis yields reliable and efﬁcient integrators for the underlying high-dimensional nonlinear evolution equations. For the mostgeneral problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicitRunge–Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by thelarger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection wecan use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of anelliptic problem. This yields a signiﬁcant gain in performance as compared to either fully implicit or explicit time integrators.
118300177
A Friendly Introduction to Numerical Analysis
(NOTE: Each chapter begins with An Overview.) 1. Getting Started. Algorithms. Convergence. Floating Point Numbers. Floating Point Arithmetic. 2. Rootfinding. Bisection Method. Method of False Position. Fixed Point Iteration. Newton's Method. The Secant Method and Muller's Method. Accelerating Convergence. Roots of Polynomials. 3. Systems of Equations. Gaussian Elimination. Pivoting Strategies. Norms. Error Estimates. LU Decomposition. Direct Factorization. Special Matrices. Iterative Techniques for Linear Systems: Basic Concepts and Methods. Iterative Techniques for Linear Systems: Conjugate-Gradient Method. Nonlinear Systems. 4. Eigenvalues and Eigenvectors. The Power Method. The Inverse Power Method. Deflation. Reduction to Tridiagonal Form. Eigenvalues of Tridiagonal and Hessenberg Matrices. 5. Interpolation and Curve Fitting. Lagrange Form of the Interpolating Polynomial. Neville's Algorithm. The Newton Form of the Interpolating Polynomial and Divided Differences. Optimal Interpolating Points. Piecewise Linear Interpolation. Hermite and Hermite Cubic Interpolation. Regression. 6. Numerical Differentiation and Integration. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations. 7. Numerical Methods for Initial Value Problems of Ordinary Differential Equations. Continuous Theory and Key Numerical Concepts. Euler's Method. Higher-Order One-Step Methods. Multistep Methods. Convergence Analysis. Error Control and Variable Step Size Algorithms. Systems of Equations and Higher-Order Equations. Absolute Stability and Stiff Equations. 8. Second-Order One-Dimensional Two-Point Boundary Value Problems. Finite Difference Method, Part I: The Linear Problem with Dirichlet Boundary Conditions. Finite Difference Method, Part II: The Linear Problem with Non-Dirichlet Boundary Conditions. Finite Difference Method, Part III: Nonlinear Problems. The Shooting Method, Part I: Linear Boundary Value Problems. The Shooting Method, Part II: Nonlinear Boundary Value Problems. 9. Finite Difference Method for Elliptic Partial Differential Equations. The Poisson Equation on a Rectangular Domain, I: Dirichlet Boundary Conditions. The Poisson Equation on a Rectangular Domain, II: Non-Dirichlet Boundary Conditions. Solving the Discrete Equations: Relaxation Schemes. Local Mode Analysis of Relaxation and the Multigrid Method. Irregular Domains. 10. Finite Difference Method for Parabolic Partial Differential Equations. The Heat Equation with Dirichlet Boundary Conditions. Stability. More General Parabolic Equations. Non-Dirichlet Boundary Conditions. Polar Coordinates. Problems in Two Space Dimensions. 11. Finite Difference Method for Hyperbolic Partial Differential Equations and the Convection-Diffusion Equation. Advection Equation, I: Upwind Differencing. Advection Equation, II: MacCormack Method. Convection-Diffusion Equation. The Wave Equation. Appendices. Appendix A. Important Theorems from Calculus. Appendix B. Algorithm for Solving a Tridiagonal System of Linear Equations. References. Index. Answers to Selected Problems.