The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and emp...The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.展开更多
With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and re...With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.展开更多
Based on the displacement discontinuity method and the discrete fracture unified pipe network model,a sequential iterative numerical method was used to build a fracturing-production integrated numerical model of shale...Based on the displacement discontinuity method and the discrete fracture unified pipe network model,a sequential iterative numerical method was used to build a fracturing-production integrated numerical model of shale gas well considering the two-phase flow of gas and water.The model accounts for the influence of natural fractures and matrix properties on the fracturing process and directly applies post-fracturing formation pressure and water saturation distribution to subsequent well shut-in and production simulation,allowing for a more accurate fracturing-production integrated simulation.The results show that the reservoir physical properties have great impacts on fracture propagation,and the reasonable prediction of formation pressure and reservoir fluid distribution after the fracturing is critical to accurately predict the gas and fluid production of the shale gas wells.Compared with the conventional method,the proposed model can more accurately simulate the water and gas production by considering the impact of fracturing on both matrix pressure and water saturation.The established model is applied to the integrated fracturing-production simulation of practical horizontal shale gas wells.The simulation results are in good agreement with the practical production data,thus verifying the accuracy of the model.展开更多
This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound cor...This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound corrector formulas for rectangular rule. Examples of numerical calculation have validated theoretical analysis.展开更多
The truncation error of improved Cotes formula is presented in this paper. It also displays an analysis on convergence order of improved Cotes formula. Examples of numerical calculation is given in the end.
This paper presents a Modified Formula for Cotes rule with fifths derivatives of endpoint and its truncation error. It also displays an analysis on convergence order of compound formula. Though compound modified formu...This paper presents a Modified Formula for Cotes rule with fifths derivatives of endpoint and its truncation error. It also displays an analysis on convergence order of compound formula. Though compound modified formula for Cotes rule with endpoint derivatives just calculates a newly-added fifths derivative of the two endpoints for each time compared with compound Cotes formula calculation, there are 2 more ranks of the convergence order in this modified formula. Examples of numerical calculation have validated theoretical analysis.展开更多
文摘The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.
文摘With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.
基金Supported by the National Natural Science Foundation of China(52374043)Key Program of the National Natural Science Foundation of China(52234003).
文摘Based on the displacement discontinuity method and the discrete fracture unified pipe network model,a sequential iterative numerical method was used to build a fracturing-production integrated numerical model of shale gas well considering the two-phase flow of gas and water.The model accounts for the influence of natural fractures and matrix properties on the fracturing process and directly applies post-fracturing formation pressure and water saturation distribution to subsequent well shut-in and production simulation,allowing for a more accurate fracturing-production integrated simulation.The results show that the reservoir physical properties have great impacts on fracture propagation,and the reasonable prediction of formation pressure and reservoir fluid distribution after the fracturing is critical to accurately predict the gas and fluid production of the shale gas wells.Compared with the conventional method,the proposed model can more accurately simulate the water and gas production by considering the impact of fracturing on both matrix pressure and water saturation.The established model is applied to the integrated fracturing-production simulation of practical horizontal shale gas wells.The simulation results are in good agreement with the practical production data,thus verifying the accuracy of the model.
文摘This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound corrector formulas for rectangular rule. Examples of numerical calculation have validated theoretical analysis.
文摘The truncation error of improved Cotes formula is presented in this paper. It also displays an analysis on convergence order of improved Cotes formula. Examples of numerical calculation is given in the end.
文摘This paper presents a Modified Formula for Cotes rule with fifths derivatives of endpoint and its truncation error. It also displays an analysis on convergence order of compound formula. Though compound modified formula for Cotes rule with endpoint derivatives just calculates a newly-added fifths derivative of the two endpoints for each time compared with compound Cotes formula calculation, there are 2 more ranks of the convergence order in this modified formula. Examples of numerical calculation have validated theoretical analysis.
