Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implic...Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this te...We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete (NAD) operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG)method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.展开更多
This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. ...This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.展开更多
The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the ...The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.展开更多
Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic ...Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.展开更多
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the nume...For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.展开更多
In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a thr...In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a three-stage stiffly accurate semi-implicit(SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.展开更多
This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix...This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.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
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
A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta met...A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.展开更多
In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and sta...In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.展开更多
Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the diffe...Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.展开更多
In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a sy...In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.展开更多
This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6<sup>th</sup> order 7 stages with the incorporated control step size in the numerical ...This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6<sup>th</sup> order 7 stages with the incorporated control step size in the numerical solution of Ordinary Differential Equations (ODE). The purpose of the present work is to construct a system of nonlinear equations and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (6,7) method (6<sup>th</sup> order and 7 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is complicated, all coefficients are found with respect to 7 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically. Some examples for five different choices of the arbitrary values of the systems are presented in this paper.展开更多
The purpose of the present work is to construct a nonlinear equation system (85 × 53) using Butcher’s Table and then by solving this system to find the values of all set parameters and finally the reduction form...The purpose of the present work is to construct a nonlinear equation system (85 × 53) using Butcher’s Table and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (7,9) method (7<sup>th</sup> order and 9 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is too complicated, we introduce a subsystem from the original system where all coefficients are found with respect to 9 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically.展开更多
This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable ...This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable multistep Runge-Kutta methods with constrained grid.The finite-dimensional and infinite-dimensional dissipativity results of-algebraically stable multistep Runge-Kutta methods are obtained.展开更多
Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are establi...Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
基金supported by ONR UMass Dartmouth Marine and UnderSea Technology(MUST)grant N00014-20-1-2849 under the project S31320000049160by DOE grant DE-SC0023164 sub-award RC114586-UMD+2 种基金by AFOSR grants FA9550-18-1-0383 and FA9550-23-1-0037supported by Michigan State University,by AFOSR grants FA9550-19-1-0281 and FA9550-18-1-0383by DOE grant DE-SC0023164.
文摘Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金This research was supported by the National Natural Science Foundation of China (Nos. 41230210 and 41204074), the Science Foundation of the Education Department of Yunnan Province (No. 2013Z152), and Statoil Company (Contract No. 4502502663).
文摘We propose a symplectic partitioned Runge-Kutta (SPRK) method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete (NAD) operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction (LWC) method and the eighth-order staggered-grid (SG)method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.
文摘This article deals with a class of numerical methods for retarded differential algebraic systems with time-variable delay. The methods can be viewed as a combination of Runge-Kutta methods and Lagrange interpolation. A new convergence concept, called DA-convergence, is introduced. The DA-convergence result for the methods is derived. At the end, a numerical example is given to verify the computational effectiveness and the theoretical result.
基金Project supported by the National Natural Science Foundation of China (No. 11071067)the Hunan Graduate Student Science and Technology Innovation Project (No. CX2011B184)
文摘The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.
基金supported by the National Natural Science Foundation of China (Nos. 10772147 and10632030)the Ph. D. Program Foundation of Ministry of Education of China (No. 20070699028)+2 种基金the Natural Science Foundation of Shaanxi Province of China (No. 2006A07)the Open Foundationof State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Foundation for Fundamental Research of Northwestern Polytechnical University
文摘Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations (PDEs) derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
文摘For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.
基金supported by the NSF(10926158) of ChinaDoctoral Fund(20090061120038) of Ministry of Education of ChinaBasic Scientific Research Foundation(200903287) of Jilin University
文摘In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations(SDEs).Two methods,a three-stage stiffly accurate semi-implicit(SASI3) method and a three-stage stiffly accurate diagonally implicit (SADI3) method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.
文摘This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.
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.
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A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.
文摘In this paper, a rather general class of explicit parallel multistep Runge-Kutta methods is constructed for solving initial value problem of ordinary differential equations. Also, the corresponding convergence and stability are analysed. Several parallel computational formulae are given. The numerical experiments, including accuracy, speedup, and efficiency tests show that the methods are efficient.
基金The NNSF (10471054) of Chinathe China Postdoctoral Science Foundationthe Youth Foundation of Institute of Mathematics, Jilin University
文摘Numerical dispersion relation of the multi-symplectic Runge-Kutta (MSRK) method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.
文摘In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.
文摘This paper is describing in detail the way we define the equations which give the formulas in the methods Runge-Kutta 6<sup>th</sup> order 7 stages with the incorporated control step size in the numerical solution of Ordinary Differential Equations (ODE). The purpose of the present work is to construct a system of nonlinear equations and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (6,7) method (6<sup>th</sup> order and 7 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is complicated, all coefficients are found with respect to 7 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically. Some examples for five different choices of the arbitrary values of the systems are presented in this paper.
文摘The purpose of the present work is to construct a nonlinear equation system (85 × 53) using Butcher’s Table and then by solving this system to find the values of all set parameters and finally the reduction formula of the Runge-Kutta (7,9) method (7<sup>th</sup> order and 9 stages) for the solution of an Ordinary Differential Equation (ODE). Since the system of high order conditions required to be solved is too complicated, we introduce a subsystem from the original system where all coefficients are found with respect to 9 free parameters. These free parameters, as well as some others in addition, are adjusted in such a way to furnish more efficient R-K methods. We use the MATLAB software to solve several of the created subsystems for the comparison of our results which have been solved analytically.
基金Inner Mongolia University 2020 undergraduate teaching reform research and construction project-NDJG2094。
文摘This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable multistep Runge-Kutta methods with constrained grid.The finite-dimensional and infinite-dimensional dissipativity results of-algebraically stable multistep Runge-Kutta methods are obtained.
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta (R-K) methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations (DAEs) in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.