We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusi...We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusing on discontinuous spectral element semidis-cretizations,we design new controllers for existing methods and for some new embedded Runge-Kutta pairs.We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice.We compare a wide range of error-control-based methods,along with the common approach in which step size con-trol is based on the Courant-Friedrichs-Lewy(CFL)number.The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances,while additionally providing control of the temporal error at tighter tolerances.The numerical examples include challenging industrial CFD applications.展开更多
In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have ...In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.展开更多
In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the l...In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.展开更多
This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods....This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.展开更多
In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators...In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.展开更多
In this paper, to begin with. the nonlinear differential equations of a truncaled shallow spherical shell with variable thickness under uniformal distributed load are linearized by step-by-step loading method. The lin...In this paper, to begin with. the nonlinear differential equations of a truncaled shallow spherical shell with variable thickness under uniformal distributed load are linearized by step-by-step loading method. The linear differential equations can be solved by spline collocanon method. Critical loads have been obtained accordingly.展开更多
针对RRT(Rapidly-exploring Random Tree)算法在机器人路径规划过程存在采样点随机性高、算法效率低、路径规划时间长以及规划路径冗长等问题,文中提出一种结合人工势场法的双向RRT路径规划算法。将传统RRT算法中单向扩展方式改为由起...针对RRT(Rapidly-exploring Random Tree)算法在机器人路径规划过程存在采样点随机性高、算法效率低、路径规划时间长以及规划路径冗长等问题,文中提出一种结合人工势场法的双向RRT路径规划算法。将传统RRT算法中单向扩展方式改为由起点和终点同时进行扩展,在节点扩展时加入人工势场法进行引导,增加节点扩展的目的性。将固定步长改换为可变步长,使随机树可以更快地向目标点扩展。对生成路径进行剪枝处理,删除路径中的冗余节点,进一步缩短路径长度。利用MATLAB仿真平台在相同环境下对比所提改进算法与RRT-Connect算法、DRRT-Connect(Dynamic Rapidly-exploring Random Tree Connect)算法、GB(Goal-Biased)-RRT算法、A^(*)算法、PRM(Probabilistic Road Map)算法的路径规划效果。仿真结果表明,所提改进算法与其他改进算法相比最短路径缩短了7%,最短搜索时间降低了65%,提高了算法的规划效率。将所提算法应用于机器人,结果证明了其具有较强可行性。展开更多
An improved finite difference method （FDM）is described to solve existing problems such as low efficiency and poor convergence performance in the traditional method adopted to derive the pressure distribution of aero...An improved finite difference method （FDM）is described to solve existing problems such as low efficiency and poor convergence performance in the traditional method adopted to derive the pressure distribution of aerostatic bearings. A detailed theoretical analysis of the pressure distribution of the orifice-compensated aerostatic journal bearing is presented. The nonlinear dimensionless Reynolds equation of the aerostatic journal bearing is solved by the finite difference method. Based on the principle of flow equilibrium, a new iterative algorithm named the variable step size successive approximation method is presented to adjust the pressure at the orifice in the iterative process and enhance the efficiency and convergence performance of the algorithm. A general program is developed to analyze the pressure distribution of the aerostatic journal bearing by Matlab tool. The results show that the improved finite difference method is highly effective, reliable, stable, and convergent. Even when very thin gas film thicknesses （less than 2 Win）are considered, the improved calculation method still yields a result and converges fast.展开更多
Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been dev...Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.展开更多
Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ord...Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ordinary differential equations(ODEs)usually found in most applied problems.This study aims to develop a new numerical method,namely the high order variable step variable order block backward differentiation formula(VSVOHOBBDF)for the main purpose of approximating the solutions of third order ODEs.The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code.The order of the proposed method was then discussed in detail.The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively.In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB,particularly ode15s and ode23s,a numerical experiment was conducted on a set of stiff problems.The numerical results prove that for this particular set of problem,the use of the proposed method is more efficient than the comparable methods.VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.展开更多
According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numeric...According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numerical method for solving differential equations is designed.The stability properties of the formulas are discussed and the stability regions are analyzed.The deduced methods are applied to a simulation problem.The results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed.展开更多
Coastal wastewater-discharged effluents contain a mixture of pollutants with decay rates that vary with water depth. Analytical models using a two-dimensional advection-diffusion equation are presented to study the ef...Coastal wastewater-discharged effluents contain a mixture of pollutants with decay rates that vary with water depth. Analytical models using a two-dimensional advection-diffusion equation are presented to study the effects of a cross-stream sudden depth change and decay on mixing and dispersing steady discharge of effluents through a sea outfall. The solutions are illustrated graphically by plotting contours of concentration, resembling snapshots of discharged effluent plumes in the far-field. Different shapes of effluent plumes are observed due to the variability of length of the step seabed, and the concentration at the step seabed is formulated to measure how much has discharged effluents dispersed into or out of the shallow coastal waters.展开更多
相较于传统的两步法,动力学一步法能充分利用观测数据的原始信息,理论上可获得更合理的时变重力场产品,同时也因其涉及的参数维度更多样、函数模型更复杂,一直是当前研究的热点和难点.本文研究并实现了动力学一步法恢复时变重力场,给出...相较于传统的两步法,动力学一步法能充分利用观测数据的原始信息,理论上可获得更合理的时变重力场产品,同时也因其涉及的参数维度更多样、函数模型更复杂,一直是当前研究的热点和难点.本文研究并实现了动力学一步法恢复时变重力场,给出了合理的数据处理策略,而后基于GRACE-FO(GRACE Follow-On)星载GPS数据和KBR(K/Ka Band Ranging)距离变率数据反演了2021—2022年60阶全球月时变重力场模型.对于一步法中诸多技术细节,本文重点分析了先验权和经验参数对轨道确定和模型反演的影响,研究表明:当采用30 s采样率的GPS数据时,需适当对GPS数据降权,以免引入过多噪声,码伪距、载波相位和KBR距离变率数据的先验权比应为1:104:1014;为了保证轨道和模型质量,在反演过程中有必要引入经验参数以吸收残余的摄动力误差,相较其他经验参数(分段周期经验加速度、几何经验参数),分段常经验加速度在保证定轨精度的同时可更有效地吸收模型中的噪声.此外,在采用相同动力学参数配置时,动力学一步法反演的时变重力场模型无论是与官方模型的一致性还是内符合精度均优于两步法.最后,综合评估了整个时间跨度的轨道和时变重力场模型质量,结果显示,动力学一步法确定的轨道可满足厘米级需求,双星的卫星激光测距残差标准差均为1.6 cm,重力场模型与官方机构CSR(Center for Space Research)、JPL(Jet Propulsion Laboratory)、GFZ(GeoForschungsZentrum Potsdam)最新发布的RL06.1模型一致性较好,在保留完整时变信号特征的前提下,噪声表现与CSR模型相当,优于JPL、GFZ模型.展开更多
We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the...We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.展开更多
基金Open Access funding enabled and organized by Projekt DEAL.
文摘We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusing on discontinuous spectral element semidis-cretizations,we design new controllers for existing methods and for some new embedded Runge-Kutta pairs.We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice.We compare a wide range of error-control-based methods,along with the common approach in which step size con-trol is based on the Courant-Friedrichs-Lewy(CFL)number.The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances,while additionally providing control of the temporal error at tighter tolerances.The numerical examples include challenging industrial CFD applications.
文摘In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.
文摘In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.
文摘This paper deals with the parallel diagonal implicit Runge-Kutta methods for solving DDEs with a constant delay. It is shown that the suitable choice of the predictor matrix can guarantee the stability of the methods. It is proved that for the suitable selection of the diagonal matrix D, the method based on Radau IIA is δ-convergent, and the estimates for the non-stiff speed and the stiff speed of convergence are given.
文摘In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.
文摘In this paper, to begin with. the nonlinear differential equations of a truncaled shallow spherical shell with variable thickness under uniformal distributed load are linearized by step-by-step loading method. The linear differential equations can be solved by spline collocanon method. Critical loads have been obtained accordingly.
文摘针对RRT(Rapidly-exploring Random Tree)算法在机器人路径规划过程存在采样点随机性高、算法效率低、路径规划时间长以及规划路径冗长等问题,文中提出一种结合人工势场法的双向RRT路径规划算法。将传统RRT算法中单向扩展方式改为由起点和终点同时进行扩展,在节点扩展时加入人工势场法进行引导,增加节点扩展的目的性。将固定步长改换为可变步长,使随机树可以更快地向目标点扩展。对生成路径进行剪枝处理,删除路径中的冗余节点,进一步缩短路径长度。利用MATLAB仿真平台在相同环境下对比所提改进算法与RRT-Connect算法、DRRT-Connect(Dynamic Rapidly-exploring Random Tree Connect)算法、GB(Goal-Biased)-RRT算法、A^(*)算法、PRM(Probabilistic Road Map)算法的路径规划效果。仿真结果表明,所提改进算法与其他改进算法相比最短路径缩短了7%,最短搜索时间降低了65%,提高了算法的规划效率。将所提算法应用于机器人,结果证明了其具有较强可行性。
基金The National Natural Science Foundation of China(No50475073,50775036)the High Technology Research Program of Jiangsu Province(NoBG2006035)
文摘An improved finite difference method （FDM）is described to solve existing problems such as low efficiency and poor convergence performance in the traditional method adopted to derive the pressure distribution of aerostatic bearings. A detailed theoretical analysis of the pressure distribution of the orifice-compensated aerostatic journal bearing is presented. The nonlinear dimensionless Reynolds equation of the aerostatic journal bearing is solved by the finite difference method. Based on the principle of flow equilibrium, a new iterative algorithm named the variable step size successive approximation method is presented to adjust the pressure at the orifice in the iterative process and enhance the efficiency and convergence performance of the algorithm. A general program is developed to analyze the pressure distribution of the aerostatic journal bearing by Matlab tool. The results show that the improved finite difference method is highly effective, reliable, stable, and convergent. Even when very thin gas film thicknesses （less than 2 Win）are considered, the improved calculation method still yields a result and converges fast.
基金supported by an NSERC Canada Postgraduate Scholarshipsupported by a grant from NSERC Canada
文摘Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.
基金funded by Fundamental Research Grant Scheme Universiti Sains Malaysia,Grant No.203/PJJAUH/6711688 received by S.A.M.Yatim.Url at http://www.research.usm.my/default.asp?tag=3&f=1&k=1.
文摘Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ordinary differential equations(ODEs)usually found in most applied problems.This study aims to develop a new numerical method,namely the high order variable step variable order block backward differentiation formula(VSVOHOBBDF)for the main purpose of approximating the solutions of third order ODEs.The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code.The order of the proposed method was then discussed in detail.The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively.In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB,particularly ode15s and ode23s,a numerical experiment was conducted on a set of stiff problems.The numerical results prove that for this particular set of problem,the use of the proposed method is more efficient than the comparable methods.VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.
基金supported by the National Natural Science Foundation of China Under Grant No.61773008.
文摘According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numerical method for solving differential equations is designed.The stability properties of the formulas are discussed and the stability regions are analyzed.The deduced methods are applied to a simulation problem.The results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed.
文摘Coastal wastewater-discharged effluents contain a mixture of pollutants with decay rates that vary with water depth. Analytical models using a two-dimensional advection-diffusion equation are presented to study the effects of a cross-stream sudden depth change and decay on mixing and dispersing steady discharge of effluents through a sea outfall. The solutions are illustrated graphically by plotting contours of concentration, resembling snapshots of discharged effluent plumes in the far-field. Different shapes of effluent plumes are observed due to the variability of length of the step seabed, and the concentration at the step seabed is formulated to measure how much has discharged effluents dispersed into or out of the shallow coastal waters.
文摘相较于传统的两步法,动力学一步法能充分利用观测数据的原始信息,理论上可获得更合理的时变重力场产品,同时也因其涉及的参数维度更多样、函数模型更复杂,一直是当前研究的热点和难点.本文研究并实现了动力学一步法恢复时变重力场,给出了合理的数据处理策略,而后基于GRACE-FO(GRACE Follow-On)星载GPS数据和KBR(K/Ka Band Ranging)距离变率数据反演了2021—2022年60阶全球月时变重力场模型.对于一步法中诸多技术细节,本文重点分析了先验权和经验参数对轨道确定和模型反演的影响,研究表明:当采用30 s采样率的GPS数据时,需适当对GPS数据降权,以免引入过多噪声,码伪距、载波相位和KBR距离变率数据的先验权比应为1:104:1014;为了保证轨道和模型质量,在反演过程中有必要引入经验参数以吸收残余的摄动力误差,相较其他经验参数(分段周期经验加速度、几何经验参数),分段常经验加速度在保证定轨精度的同时可更有效地吸收模型中的噪声.此外,在采用相同动力学参数配置时,动力学一步法反演的时变重力场模型无论是与官方模型的一致性还是内符合精度均优于两步法.最后,综合评估了整个时间跨度的轨道和时变重力场模型质量,结果显示,动力学一步法确定的轨道可满足厘米级需求,双星的卫星激光测距残差标准差均为1.6 cm,重力场模型与官方机构CSR(Center for Space Research)、JPL(Jet Propulsion Laboratory)、GFZ(GeoForschungsZentrum Potsdam)最新发布的RL06.1模型一致性较好,在保留完整时变信号特征的前提下,噪声表现与CSR模型相当,优于JPL、GFZ模型.
基金Open Access funding enabled and organized by Projekt DEAL.Andrew Winters was funded through Vetenskapsrådet,Sweden Grant Agreement 2020-03642 VR.Some computations were enabled by resources provided by the Swedish National Infrastructure for Computing(SNIC)at Tetralith,par-tially funded by the Swedish Research Council under Grant Agreement No.2018-05973Hugo Guillermo Castro was funded through the award P2021-0004 of King Abdullah University of Science and Technol-ogy.Some of the simulations were enabled by the Supercomputing Laboratory and the Extreme Comput-ing Research Center at King Abdullah University of Science and Technology.Gregor Gassner acknowl-edges funding through the Klaus-Tschira Stiftung via the project“HiFiLab”.Gregor Gassner and Michael Schlottke-Lakemper acknowledge funding from the Deutsche Forschungsgemeinschaft through the research unit“SNuBIC”(DFG-FOR5409).
文摘We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fuid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations.We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)number.Our numerical examples show that the error-based step size control is easy to use,robust,and efcient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy projections.We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.
