In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two ...In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.展开更多
Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations submit various critical physical phenomena with a typical equation for optical fibres with ...Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations. Many methods used to solve coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.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
It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the...It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the order of an s-stage monoimplicit Runge-Kutta method is at most s+1 and the stage order is at most 3. In this paper, it is shown that the order of an s-stage mono-implicit Runge-Kutta method being algebraically stable is at most min((s) over tilde, 4), and the stage order together with the optimal B-convergence order is at most min(s, 2), where [GRAPHICS]展开更多
The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p + l(1 less than or equal to p...The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p + l(1 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 0, 1, 2. (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 < p <less than or equal to> s. Then this method is symplectic if and only if either p = s or the nonlinear stability matrix M of the method has an (s - p) x (s - p) chief submatrix (M) over cap = 0. (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has been designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying C(p),D(p) and B(2p + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.展开更多
This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to so...This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation laws.The application discussed here is the solution of 3-D problems on unstructured meshes.Our numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.展开更多
We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respe...We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respectively,are considered in this paper.Stochastic Runge-Kutta(SRK)methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplectic property are given.Based on the weak/strong order and symplectic conditions,some effective schemes are derived.In particular,using the algebraic computation,we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise,and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise,respectively.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.展开更多
The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and cal...The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.展开更多
In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a syn...In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a synergetic model of electricity market operation system, and studied the dynamic process of the system with empirical example, which revealed the internal mechanism of the system evolution. In order to verify the accuracy of the synergetic model, fourth-order Runge-Kutta algorithm and grey relevance method were used. Finally, we found that the reserve rate of generation was the order parameter of the system. Then we can use the principle of Synergetics to evaluate the efficiency of electricity market operation.展开更多
In the current work, we study two infectious disease models and we use nonlinear optimization and optimal control theory which helps to find strategies towards transmission control and to forecast the international sp...In the current work, we study two infectious disease models and we use nonlinear optimization and optimal control theory which helps to find strategies towards transmission control and to forecast the international spread of the infectious diseases. The relationship between epidemiology, mathematical modeling and computational tools lets us to build and test theories on the development and fighting with a disease. This study is motivated by the study of epidemiological models applied to infectious diseases in an optimal control perspective. We use the numerical methods to display the solutions of the optimal control problems to find the effect of vaccination on these models. Finally, global sensitivity analysis LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC) has been performed to investigate the key parameters in model equations. This present work will advance the understanding about the spread of infectious diseases and lead to novel conceptual understanding for spread of them.展开更多
In this paper, simulation of InAs/GaAs quantum dot (QD) laser is performed based upon a set of eight rate equations for the carriers and photons in five energy states. Carrier dynamics in these lasers were under analy...In this paper, simulation of InAs/GaAs quantum dot (QD) laser is performed based upon a set of eight rate equations for the carriers and photons in five energy states. Carrier dynamics in these lasers were under analysis and the rate equations are solved using 4th order Runge-Kutta method. We have shown that by increasing injected current to the active medium of laser, switching-on and stability time of the system would decrease and power peak and stationary power will be increased. Also, emission in any state will start when the lower state is saturated and remain steady. The results including P-I characteristic curve for the ground state (GS), first excited state (ES1), second excited state (ES2) and output power of the QD laser will be presented.展开更多
文摘In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.
文摘Models of the coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations. Many methods used to solve coupled nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.
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It is well known that mono-implicit Runge-Kutta methods have been applied in the efficient numerical solution of initial or boundary value problems of ordinary differential equations. Burrage (1994) has shown that the order of an s-stage monoimplicit Runge-Kutta method is at most s+1 and the stage order is at most 3. In this paper, it is shown that the order of an s-stage mono-implicit Runge-Kutta method being algebraically stable is at most min((s) over tilde, 4), and the stage order together with the optimal B-convergence order is at most min(s, 2), where [GRAPHICS]
基金the National Natural Science Foundation of China.
文摘The main results of this paper are as follows: (1) Suppose an s stage Runge-Kutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p + l(1 less than or equal to p less than or equal to s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where 1 0, 1, 2. (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 < p <less than or equal to> s. Then this method is symplectic if and only if either p = s or the nonlinear stability matrix M of the method has an (s - p) x (s - p) chief submatrix (M) over cap = 0. (3) Using the results (1) and (2) as bases, we present a general approach for the construction of symplectic Runge-Kutta methods, and a software has been designed, by means of which, the coefficients of s stage symplectic Runge-Kutta methods satisfying C(p),D(p) and B(2p + l) can be easily computed, where 1 I P less than or equal to s, 0 less than or equal to l less than or equal to 2, s less than or equal to 2p + l less than or equal to 2s.
基金The research was partially supported by NSFC grant 10931004,10871093,11002071 and the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simulations.
文摘This paper further considers weighted essentially non-oscillatory(WENO)and Hermite weighted essentially non-oscillatory(HWENO)finite volume methods as limiters for Runge-Kutta discontinuous Galerkin(RKDG)methods to solve problems involving nonlinear hyperbolic conservation laws.The application discussed here is the solution of 3-D problems on unstructured meshes.Our numerical tests again demonstrate this is a robust and high order limiting procedure,which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.
基金This work was supported by NSFC(91130003)The first authors is also supported by NSFC(11101184,11271151)+1 种基金the Science Foundation for Young Scientists of Jilin Province(20130522101JH)The second and third authors are also supported by NSFC(11021101,11290142).The authors would like to thank anonymous reviewers for careful reading and invaluable suggestions,which greatly improved the presentation of the paper.
文摘We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respectively,are considered in this paper.Stochastic Runge-Kutta(SRK)methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplectic property are given.Based on the weak/strong order and symplectic conditions,some effective schemes are derived.In particular,using the algebraic computation,we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise,and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise,respectively.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
基金support from the National Natural Science Foundation of China(NSFC)through its grant 51377098.
文摘The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.
文摘In Synergetics, when a complex system evolves from one sate to another, the order parameter plays a dominant role. We can analyze the complex system state by studying the dynamic of order parameter. We developed a synergetic model of electricity market operation system, and studied the dynamic process of the system with empirical example, which revealed the internal mechanism of the system evolution. In order to verify the accuracy of the synergetic model, fourth-order Runge-Kutta algorithm and grey relevance method were used. Finally, we found that the reserve rate of generation was the order parameter of the system. Then we can use the principle of Synergetics to evaluate the efficiency of electricity market operation.
文摘In the current work, we study two infectious disease models and we use nonlinear optimization and optimal control theory which helps to find strategies towards transmission control and to forecast the international spread of the infectious diseases. The relationship between epidemiology, mathematical modeling and computational tools lets us to build and test theories on the development and fighting with a disease. This study is motivated by the study of epidemiological models applied to infectious diseases in an optimal control perspective. We use the numerical methods to display the solutions of the optimal control problems to find the effect of vaccination on these models. Finally, global sensitivity analysis LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC) has been performed to investigate the key parameters in model equations. This present work will advance the understanding about the spread of infectious diseases and lead to novel conceptual understanding for spread of them.
文摘In this paper, simulation of InAs/GaAs quantum dot (QD) laser is performed based upon a set of eight rate equations for the carriers and photons in five energy states. Carrier dynamics in these lasers were under analysis and the rate equations are solved using 4th order Runge-Kutta method. We have shown that by increasing injected current to the active medium of laser, switching-on and stability time of the system would decrease and power peak and stationary power will be increased. Also, emission in any state will start when the lower state is saturated and remain steady. The results including P-I characteristic curve for the ground state (GS), first excited state (ES1), second excited state (ES2) and output power of the QD laser will be presented.
