Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for...Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDE; Review of the related concepts and results on RK methods; Information on the asymptotic stability and global stability of the induced NRK method.展开更多
B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-...B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.展开更多
Presents a study which derived a way of constructing symplectic methods with the help of symplecticity conditions of partitioned Runge-Kutta methods. Classes of symplectic Runge-Kutta methods; Relationship between Run...Presents a study which derived a way of constructing symplectic methods with the help of symplecticity conditions of partitioned Runge-Kutta methods. Classes of symplectic Runge-Kutta methods; Relationship between Runge-Kutta methods.展开更多
Microseismic(MS)event locations are vital aspect of MS monitoring technology used to delineate the damage zone inside the surrounding rock mass.However,complex geological conditions can impose significantly adverse ef...Microseismic(MS)event locations are vital aspect of MS monitoring technology used to delineate the damage zone inside the surrounding rock mass.However,complex geological conditions can impose significantly adverse effects on the final location results.To achieve a high-accuracy location in a complex cavern-containing structure,this study develops an MS location method using the fast marching method(FMM)with a second-order difference approach(FMM2).Based on the established velocity model with three-dimensional(3D)discrete grids,the realization of the MS location can be achieved by searching the minimum residual between the theoretical and actual first arrival times.Moreover,based on the calculation results of FMM2,the propagation paths from the MS sources to MS sensors can be obtained using the linear interpolation approach and the Runge–Kutta method.These methods were validated through a series of numerical experiments.In addition,our proposed method was applied to locate the recorded blasting and MS events that occurred during the excavation period of the underground caverns at the Houziyan hydropower station.The location results of the blasting activities show that our method can effectively reduce the location error compared with the results based on the uniform velocity model.Furthermore,the obtained MS location was verified through the occurrence of shotcrete fractures and spalling,and the monitoring results of the in-situ multipoint extensometer.Our proposed method can offer a more accurate rock fracture location and facilitate the delineation of damage zones inside the surrounding rock mass.展开更多
Efficient numerical schemes were presented for the steady state solutions of towed marine cables. For most of towed systems,the steady state problem can be resolved into two-point boundary-value problem,or initial val...Efficient numerical schemes were presented for the steady state solutions of towed marine cables. For most of towed systems,the steady state problem can be resolved into two-point boundary-value problem,or initial value problem in some special cases where the initial values are available directly.A new technique was proposed and attempted to solve the two-point boundary-value problem rather than the conventional shooting method due to its algorithm complexity and low efficiency.First,the boundary conditions are transformed into a set of nonlinear governing equations about the initial values,then bisection method is employed to solve these nonlinear equations with the aid of 4th order Runge-Kutta method.In common sense,non-uniform (sheared) current is assumed,which varies in magnitude and direction with depth.The schemes are validated through the DE Zoysa's example,then several numerical examples are also presented to illustrate the numerical schemes.展开更多
Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numer...Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.展开更多
Focuses on a study which examined the numerical solution of delay differential equations (DDE). Information on the Runge-Kutta methods for DDE; Results of the D-convergence analysis of Runge-Kutta methods; Details on ...Focuses on a study which examined the numerical solution of delay differential equations (DDE). Information on the Runge-Kutta methods for DDE; Results of the D-convergence analysis of Runge-Kutta methods; Details on the equations with several delays.展开更多
An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the appr...An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the approximate solution obtained by former researchers.展开更多
The original ghost fluid method (GFM) developed in [13] and the modifiedGFM (MGFM) in [26] have provided a simple and yet flexible way to treat twomediumflow problems. The original GFM and MGFM make the material inter...The original ghost fluid method (GFM) developed in [13] and the modifiedGFM (MGFM) in [26] have provided a simple and yet flexible way to treat twomediumflow problems. The original GFM and MGFM make the material interface"invisible" during computations and the calculations are carried out as for a singlemedium such that its extension to multi-dimensions becomes fairly straightforward.The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservationlaws is a high order accurate finite element method employing the usefulfeatures from high resolution finite volume schemes, such as the exact or approximateRiemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,we investigate using RKDG finite element methods for two-medium flow simulationsin one and two dimensions in which the moving material interfaces is treated via nonconservativemethods based on the original GFM and MGFM. Numerical results forboth gas-gas and gas-water flows are provided to show the characteristic behaviors ofthese combinations.展开更多
文摘Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDE; Review of the related concepts and results on RK methods; Information on the asymptotic stability and global stability of the induced NRK method.
文摘B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.
文摘Presents a study which derived a way of constructing symplectic methods with the help of symplecticity conditions of partitioned Runge-Kutta methods. Classes of symplectic Runge-Kutta methods; Relationship between Runge-Kutta methods.
基金the Key Program of National Natural Science Foundation of China(52039007)for providing financial support.
文摘Microseismic(MS)event locations are vital aspect of MS monitoring technology used to delineate the damage zone inside the surrounding rock mass.However,complex geological conditions can impose significantly adverse effects on the final location results.To achieve a high-accuracy location in a complex cavern-containing structure,this study develops an MS location method using the fast marching method(FMM)with a second-order difference approach(FMM2).Based on the established velocity model with three-dimensional(3D)discrete grids,the realization of the MS location can be achieved by searching the minimum residual between the theoretical and actual first arrival times.Moreover,based on the calculation results of FMM2,the propagation paths from the MS sources to MS sensors can be obtained using the linear interpolation approach and the Runge–Kutta method.These methods were validated through a series of numerical experiments.In addition,our proposed method was applied to locate the recorded blasting and MS events that occurred during the excavation period of the underground caverns at the Houziyan hydropower station.The location results of the blasting activities show that our method can effectively reduce the location error compared with the results based on the uniform velocity model.Furthermore,the obtained MS location was verified through the occurrence of shotcrete fractures and spalling,and the monitoring results of the in-situ multipoint extensometer.Our proposed method can offer a more accurate rock fracture location and facilitate the delineation of damage zones inside the surrounding rock mass.
文摘Efficient numerical schemes were presented for the steady state solutions of towed marine cables. For most of towed systems,the steady state problem can be resolved into two-point boundary-value problem,or initial value problem in some special cases where the initial values are available directly.A new technique was proposed and attempted to solve the two-point boundary-value problem rather than the conventional shooting method due to its algorithm complexity and low efficiency.First,the boundary conditions are transformed into a set of nonlinear governing equations about the initial values,then bisection method is employed to solve these nonlinear equations with the aid of 4th order Runge-Kutta method.In common sense,non-uniform (sheared) current is assumed,which varies in magnitude and direction with depth.The schemes are validated through the DE Zoysa's example,then several numerical examples are also presented to illustrate the numerical schemes.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10375039 and 90503008)the Doctoral Program Foundation from the Ministry of Education of China,and the Center of Nuclear Physics of HIRFL of China
文摘Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm.
基金NSF of China (No.19871070) and China Postdoctoral Science Foundation.
文摘Focuses on a study which examined the numerical solution of delay differential equations (DDE). Information on the Runge-Kutta methods for DDE; Results of the D-convergence analysis of Runge-Kutta methods; Details on the equations with several delays.
基金The work is financially supported by the National Natural Science Foundations of China (No.50476083).
文摘An improved shooting method was presented for solving the natural convention boundary layer equations, with a coupling of the velocity field to the temperature field. The numerical results are consistent with the approximate solution obtained by former researchers.
基金NSFC grant 10671091Nanjing University Talent Development Foundation and SRF for ROCS,SEM.Additional support was provided by NUS Research Project R-265-000-118-112 while he was in residence at the Department of Mechanical Engineering,National University of Singapore,Singapore 119260.
文摘The original ghost fluid method (GFM) developed in [13] and the modifiedGFM (MGFM) in [26] have provided a simple and yet flexible way to treat twomediumflow problems. The original GFM and MGFM make the material interface"invisible" during computations and the calculations are carried out as for a singlemedium such that its extension to multi-dimensions becomes fairly straightforward.The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservationlaws is a high order accurate finite element method employing the usefulfeatures from high resolution finite volume schemes, such as the exact or approximateRiemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper,we investigate using RKDG finite element methods for two-medium flow simulationsin one and two dimensions in which the moving material interfaces is treated via nonconservativemethods based on the original GFM and MGFM. Numerical results forboth gas-gas and gas-water flows are provided to show the characteristic behaviors ofthese combinations.
