In this paper, a special three-step difference scheme is applied to the solution of nonlinear time-evolution equations, whose coefficients are determined according to accuracy constraints, necessary conditions of squa...In this paper, a special three-step difference scheme is applied to the solution of nonlinear time-evolution equations, whose coefficients are determined according to accuracy constraints, necessary conditions of square conservation, and historical observation information under the linear supposition. As in the linear case, the schemes also have obvious superiority in overall performance in the nonlinear case compared with traditional finite difference schemes, e.g., the leapfrog(LF) scheme and the complete square conservation difference(CSCD) scheme that do not use historical observations in determining their coefficients, and the retrospective time integration(RTI) scheme that does not consider compatibility and square conservation. Ideal numerical experiments using the one-dimensional nonlinear advection equation with an exact solution show that this three-step scheme minimizes its root mean square error(RMSE) during the first 2500 integration steps when no shock waves occur in the exact solution, while the RTI scheme outperforms the LF scheme and CSCD scheme only in the first 1000 steps and then becomes the worst in terms of RMSE up to the 2500th step. It is concluded that reasonable consideration of accuracy, square conservation, and historical observations is also critical for good performance of a finite difference scheme for solving nonlinear equations.展开更多
The Preissmann implicit scheme was used to discretize the one-dimensional Saint-Venant equations, the river-junction-fiver method was applied to resolve the hydrodynamic and water quality model for river networks, and...The Preissmann implicit scheme was used to discretize the one-dimensional Saint-Venant equations, the river-junction-fiver method was applied to resolve the hydrodynamic and water quality model for river networks, and the key issues on the model were expatiated particularly in this article. This water quality module was designed to compute time dependent concentrations of a series of constituents, which are primarily governed by the processes of advection, dispersion and chemical reactions. Based on the theory of Water Quality Analysis Simulation Program （WASP） water quality model, emphasis was given to the simulation of the biogeochemical transformations that determine the fate of nutrients, in particular, the simulation of the aquatic cycles of nitrogen and phosphorus compounds. This model also includes procedures for the determination of growth and death of phytoplankton. This hydrodynamic and water quality model was applied to calculate two river networks. As illustrated by the numerical examples, the calculated water level and discharge agree with the measured data and the simulated trends and magnitudes of water quality constituents are generally in good agreement with field observations. It is concluded that the presented model is useful in the pollutant control and in the determination of pollutant-related problems for river networks.展开更多
基金the Ministry of Science and Technology of China for the National Basic Research Program of China(973 Program,Grant No.2011CB309704)
文摘In this paper, a special three-step difference scheme is applied to the solution of nonlinear time-evolution equations, whose coefficients are determined according to accuracy constraints, necessary conditions of square conservation, and historical observation information under the linear supposition. As in the linear case, the schemes also have obvious superiority in overall performance in the nonlinear case compared with traditional finite difference schemes, e.g., the leapfrog(LF) scheme and the complete square conservation difference(CSCD) scheme that do not use historical observations in determining their coefficients, and the retrospective time integration(RTI) scheme that does not consider compatibility and square conservation. Ideal numerical experiments using the one-dimensional nonlinear advection equation with an exact solution show that this three-step scheme minimizes its root mean square error(RMSE) during the first 2500 integration steps when no shock waves occur in the exact solution, while the RTI scheme outperforms the LF scheme and CSCD scheme only in the first 1000 steps and then becomes the worst in terms of RMSE up to the 2500th step. It is concluded that reasonable consideration of accuracy, square conservation, and historical observations is also critical for good performance of a finite difference scheme for solving nonlinear equations.
基金Project supported by the National Natural Science Foundation of China (Grant No.50839001)the National Basic Research Program of China (973 Program, Grant No. 2005CB724202).
文摘The Preissmann implicit scheme was used to discretize the one-dimensional Saint-Venant equations, the river-junction-fiver method was applied to resolve the hydrodynamic and water quality model for river networks, and the key issues on the model were expatiated particularly in this article. This water quality module was designed to compute time dependent concentrations of a series of constituents, which are primarily governed by the processes of advection, dispersion and chemical reactions. Based on the theory of Water Quality Analysis Simulation Program （WASP） water quality model, emphasis was given to the simulation of the biogeochemical transformations that determine the fate of nutrients, in particular, the simulation of the aquatic cycles of nitrogen and phosphorus compounds. This model also includes procedures for the determination of growth and death of phytoplankton. This hydrodynamic and water quality model was applied to calculate two river networks. As illustrated by the numerical examples, the calculated water level and discharge agree with the measured data and the simulated trends and magnitudes of water quality constituents are generally in good agreement with field observations. It is concluded that the presented model is useful in the pollutant control and in the determination of pollutant-related problems for river networks.
