Traditional Runge-Kutta Methods Research Articles

Learning Topological Horseshoe via Deep Neural Networks

Deep Neural Networks (DNNs) have been successfully applied to investigations of numerical dynamics of finite-dimensional nonlinear systems such as ODEs instead of finding numerical solutions to ODEs via the traditional Runge–Kutta method and its variants. To show the advantages of DNNs, in this paper, we demonstrate that the DNNs are more efficient in finding topological horseshoes in chaotic dynamical systems.

Improved Runge-Kutta Method for Oscillatory Problem Solution Using Trigonometric Fitting Approach

This paper provides a four-stage Trigonometrically Fitted Improved Runge-Kutta (TFIRK4) method of four orders to solve oscillatory problems, which contains an oscillatory character in the solutions. Compared to the traditional Runge-Kutta method, the Improved Runge-Kutta (IRK) method is a natural two-step method requiring fewer steps. The suggested method extends the fourth-order Improved Runge-Kutta (IRK4) method with trigonometric calculations. This approach is intended to integrate problems with particular initial value problems (IVPs) using the set functions and for trigonometrically fitted. To improve the method's accuracy, the problem primary frequency is used. The novel method is more accurate than the conventional Runge-Kutta method and IRK4. Several test problems for the system of first-order ordinary differential equations carry out numerically to demonstrate the effectiveness of this approach. The computational studies show that the TFIRK4 approach is more efficient than the existing Runge-Kutta methods.

On the application of the double integral method with quadratic temperature profile for spherical solidification of lead and tin metals

• High Stefan and Biot solidification problem considered and modelled. • The proposed method achieved a reduction of 92.73% compared to previous approaches. • Gaussian Quadrature is 3 orders of magnitude less expensive compared to Runge-Kutta. In this paper we developed the double integral method to solve the solidification problem of liquid tin and lead metals in spherical geometry with a high Stefan and Biot numbers. The solidification time solutions were obtained by deriving the heat conduction equation using a quadratic temperature profile. The set of equations were solved using a Gaussian Quadrature approach; a novelty for this problem. Hence, we intend to improve the numerical accuracy against the traditional simple integral methods in the literature. Overall, the double integral method with the quadratic profile performed better than the single integral method with a quadratic profile; for Bi = 10 and Ste = 2, the reduction in solidification time relative error using the double integral method was 92,73% compared to the simple integral method. Compared to the experimental data of lead and tin, the double integral method performed better than the single integral method for all Biot and Stefan conditions; for tin cooled by water, the double integral method was 20,06% more accurate in predicting the solidification time compared to the single integral method. Furthermore, we evaluated that using a Gaussian Quadrature integrator led to a reduction in computational cost up to 3 orders of magnitude compared to traditional Runge-Kutta methods. Therefore, our studies attested to the correctness and suitability of the developed double integral method for applications in the solidification of metals.

High order sign-preserving and well-balanced exponential Runge-Kutta discontinuous Galerkin methods for the shallow water equations with friction

In this paper, we propose a family of second and third order temporal integration methods for systems of stiff ordinary differential equations, and explore their application in solving the shallow water equations with friction. The new temporal discretization methods come from a combination of the traditional Runge-Kutta method (for non-stiff equation) and exponential Runge-Kutta method (for stiff equation), and are shown to have both the sign-preserving and steady-state-preserving properties. They are combined with the well-balanced discontinuous Galerkin spatial discretization to solve the nonlinear shallow water equations with non-flat bottom topography and (stiff) friction terms. We have demonstrated that the fully discrete schemes satisfy the well-balanced, positivity-preserving and sign-preserving properties simultaneously. The proposed methods have been tested and validated on several one- and two-dimensional test cases, and good numerical results have been observed.

Initial Value Methods for the Numerical Simulation of Fuzzy Two-Point Boundary Value Problems Using General Linear Method

This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP) using general linear method (GLM). The author derived the method, which is a combination of a Runge-Kutta type method and multi-step method. It is originally designed to solve initial value problems. It requires fewer function evaluations than the traditional Runge-Kutta methods making it computationally more efficient in achieving the required accuracy. The author will utilize the combination of the GLM with initial value methods to solve the linear fuzzy BVP's and a shooting-like method for the nonlinear cases. Numerical testing and simulation of several examples, considered by other authors, will be presented to show the efficiency of the proposed method.

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral-like resolution will be presented. Based on the high-order gas evolution model, both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. This strategy needs time accurate solution at a cell interface, which cannot be achieved by the Riemann problem based flow solvers, even though they can also provide the interface flux functions and interface flow variables. Instead, in order to update the slopes in the Riemann-solver based schemes, such as HWENO, there are additional governing equations for slopes or equivalent degrees of freedom inside each cell. In GKS, only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables. Based on both cell averaged values and their slopes, compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed. As analyzed in this paper, the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives. For nonlinear gas dynamic evolution, in order to avoid spurious oscillation in discontinuous region, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation, which covers a physical process from an initial non-equilibrium state to a final equilibrium one. Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction, and the equilibrium state is basically constructed from the linear symmetric reconstruction, the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function. This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number ≥ 0.3, has the robustness as a 2nd-order scheme, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique. At the same time, the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface. Due to the use of multi-stage multi-derivative (MSMD) time-stepping technique, for achieving a 4th-order time accuracy, the current scheme uses only two stages instead of four in the traditional Runge-Kutta method. As a result, the current GKS becomes much more efficient than the corresponding same order DG method. A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS. The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions, i.e., strong shock discontinuity, shear instability, aero-acoustic wave propagation, and NS solutions. It promotes the development of high-order scheme to a new level of maturity. The success of the current scheme crucially depends on the high-order gas evolution model, which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.

Explicit solution of gas film pressure for performance calculation of spiral grooved gas seals

The study presents an explicit solution to a narrow groove theory for compressible spiral grooved gas seals. The explicit solution was derived from Muijderman’s narrow groove theory by employing Cauchy’s integral theorem. It expresses the gas film pressure as a function of gas film radius. Using this process, the initial value problem of Muijderman’s equations is avoided, and the gas film pressure can be obtained directly in contrast to the traditional Runge-Kutta method. A complete analytical procedure of the explicit solution is presented. The accuracy of the explicit solution was validated using published results.

An Approximate Solution of Muijderman's Model for Performance Calculation of Spiral Grooved Gas Seal

This paper presents an approximate solution of Muijderman's model for compressible spiral grooved gas film. The approximate solution is derived from Muijderman's equations by Adomian decomposition method. The obtained approximate solution expresses the gas film pressure as a function of the gas film radius. The traditional Runge–Kutta method is avoided. The accuracy of the approximate solution is acceptable, and it brings convenience for performance calculation of spiral grooved gas seal. A complete Adomian decomposition procedure of Muijderman's equations is presented. The approximate solution is validated with published results.

Dynamic behavior of sail tower SPS based on the symplectic Runge-Kutta method

Space-based solar power as a promising approach to achieve clean and renewable energy had received extensive attention during the last several decades. The dynamics and stability of system of solar power satellite (SPS) in orbit were investigated through the application of Hamilton theory. First, the simplified structure model of sail tower SPS was built, and dynamic governing equation of simplified model of sail tower SPS in the Hamilton form were derived by introducing the concepts of generalized coordinates and generalized momentums and implementing variation method. Then, by numerically solving the equation through the application of the symplectic Runge-Kutta method and comparing the results with the traditional Runge-Kutta method, numerical examples demonstrated that the results achieved by the symplectic Runge-Kutta method indicated the energy preservation as well as the vibrational amplitude of the tether. Finally, the equilibrium points was calculated based on mechanical balance principle. Besides, the stability of the considered system at the equilibrium points was analyzed via numerical examples.

Higher Order Variational Integrators for Multibody System Dynamics with Constraints

The continuous and discrete Euler-Lagrangian equations with holonomic constraints are presented based on continuous and discrete Hamiltonian Principle. Using Lagrangian polynomial to interpolate state variables and Gauss quadrature formula to approximate Hamiltonian action integral, the higher order variational Galerkin integrators for multibody system dynamics with constraints and the computation procedure are given. Numerical examples are provided to show the long-time behavior of the methods proposed in this paper via comparisons with traditional Runge-Kutta methods.

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.

Singular optimal control for stochastic linear quadratic singular system using ant colony programming

In this article, singular optimal control for stochastic linear singular system with quadratic performance is obtained using ant colony programming (ACP). To obtain the optimal control, the solution of matrix Riccati differential equation is computed by solving differential algebraic equation using a novel and nontraditional ACP approach. The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution computed by the ACP approach to the problem is qualitatively better. The solution of this novel method is compared with the traditional Runge Kutta method. An illustrative numerical example is presented for the proposed method.

Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular fuzzy system with cross-term using neural networks

In this paper, solution of generalized matrix Riccati differential equation (GMRDE) for indefinite stochastic linear quadratic singular fuzzy system with cross-term is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of GMRDE obtained from well-known traditional Runge Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of GMRDE is computed by feed forward neural network (FFNN). Accuracy of the solution of the neural network approach to this problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

Solution of matrix Riccati differential equation for nonlinear singular system using neural networks

ABSTRACT In this paper, the solution of the matrix Riccati differential equation(MRDE) for nonlinear singular system is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the MRDE obtained from well known traditional Runge Kutta(RK)method and nontraditional neural network method. Accuracy of the neural solution to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method. Key words: Matrix Riccati differential equation, Nonlinear, Optimal control, Singular system, Runge Kutta method and Neural networks. 1. INTRODUCTION Neural networks or simply neural nets are computing systems, which can be trained to learn a complex relationship between two or many variables or data sets. Having the structures similar to their biological counterparts, neural networks are representational and computational models processing information in a parallel distributed fashion composed of interconnecting simple processing nodes [36]. Neural net techniques have been successfully applied in various fields such as function approximation, signal processing and adaptive (or) learning control for nonlinear systems. Using neural networks, a variety of off-line learning control algorithms have been developed for nonlinear systems [17, 25]. A variety of numerical algorithms have been developed for solving the algebraic Riccati equation. In recent years, neural network problems have attracted considerable attention of many researchers for numerical aspects for algebraic Riccati equations see [13, 14, 37, 3]. Singular systems contain a mixture of algebraic and differential equations. In that sense, the algebraic equations represent the constraints to the solution of the differential part. These systems are also known as degenerate, descriptor or semi-state and generalized state-space systems. The complex nature of singular system causes many difficulties in the analytical and numerical treatment of such systems, particularly when there is a need for their control. The system arises naturally as a linear approximation of system models or linear system models in many applications such as electrical networks, aircraft dynamics, neutral delay systems, chemical, thermal and diffusion processes, large-scale systems, robotics, biology, etc., see [6, 7, 11, 19]. Most of the research on nonlinear singular systems has focused primarily on issues related to solvability and numerical solutions for such systems [5, 9]. The literature on feedback control of nonlinear singular systems is sparse. The feedback stabilization problem for nonlinear singular systems is addressed by McClamroch [22]. In this paper, we make use of a result that generalizes the LQ theory to nonlinear systems to provide a nonlinear design method [4, 21]. This nonlinear quadratic (NLQ) method applies to systems having a broad class of nonlinear dynamics with state dependent weighting matrices. In brief, it turns out that the infinite time horizon LQ regulator problem, when solved afresh at every point on the state trajectory, leads to an asymptotically optimal control policy. The LQ regulator problem converges to the optimal control close to the origin. As the theory of optimal control of linear systems with quadratic performance criteria is well developed, the results are most complete and close to use in many practical designing problems. The theory of the quadratic cost control problem has been treated as a more interesting problem and the optimal feedback with minimum cost control has been characterized by the solution of a Riccati equation. Da Prato and Ichikawa [12] showed that the optimal feedback control and the minimum cost are characterized by the solution of a Riccati equation. Solving the MRDE is the central issue in optimal control theory. The needs for solving such equations often arise in analysis and synthesis such as

Optimal control for stochastic nonlinear singular system using neural networks

In this paper, optimal control for stochastic nonlinear singular system with quadratic performance is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix Riccati differential equation (MRDE) obtained from the well-known traditional Runge–Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of MRDE is computed by feedforward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

Optimal control for stochastic linear quadratic singular system using neural networks

In this paper, optimal control for stochastic linear singular system with quadratic performance is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix Riccati differential equation (MRDE) obtained from well known traditional Runge–Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of MRDE is computed by feed forward neural network (FFNN). Accuracy of the solution of the neural network approach to the problem is qualitatively better. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

Optimal control for linear system using genetic programming

AbstractIn this paper, optimal control for a linear system with quadratic performance is obtained using genetic programming (GP). The goal is to find the optimal control with reduced calculus effort using non‐traditional methods. The obtained GP solution is compared with the traditional Runge–Kutta method. To obtain optimal control, the solution of matrix Riccati differential equation is computed based on grammatical evolution. The accuracy of the solution of the GP approach to the problem is qualitatively better than traditional methods. An illustrative numerical example is presented for the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular system using neural networks

In this paper, solution of generalized matrix Riccati differential equation (GMRDE) for indefinite stochastic linear quadratic singular system is obtained using neural networks. The goal is to provide optimal control with reduced calculus effort by comparing the solutions of GMRDE obtained from well known traditional Runge Kutta (RK) method and nontraditional neural network method. To obtain the optimal control, the solution of GMRDE is computed by feed forward neural network (FFNN). Accuracy of the solution of the neural network approach to the problem is qualitatively better. The neural network solution is also compared with the solution of ode45, a standard solver available in MATLAB which implements RK method for variable step size. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

Optimal control for nonlinear singular systems with quadratic performance using neural networks

In this paper, optimal control for nonlinear singular system with quadratic performance is obtained using neural networks. This control is in the form of a state feedback function plus a time function. Furthermore, the main feature is that optimality is always preserved by this control; hence, this control is called a quasi-feedback optimal control. To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is obtained by feedforward neural network (FFNN). Accuracy of the neural network solution to the optimal control problem is qualitatively better than Runge–Kutta (RK) solution. The advantage of the proposed approach is that, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. A numerical example is presented to illustrate the implementation of artificial neural networks (ANN) to obtain optimal control.

Fast and Effective Analysis Methods for Fiber Raman Amplifiers

A series of new single-step methods and their corresponding algorithms with automatic step size adjustment for model equations of fiber Raman amplifiers are proposed and compared in this paper. On the basis of the Newton–Raphson method, multiple shooting algorithms for the two-point boundary value problems involved in solving Raman amplifier propagation equations are constructed. A verified example shows that, compared with the traditional Runge–Kutta methods, the proposed methods can increase the accuracy by more than two orders of magnitude under the same conditions. The simulations for Raman amplifier propagation equations demonstrate that our methods can increase the computing speed by more than 5 times, extend the step size significantly, and improve the stability in comparison with the Dormand–Prince method. The numerical results show that the combination of the multiple shooting algorithms and the proposed methods has the capacity to rapidly and effectively solve the model equations of multipump Raman amplifiers under various conditions such as co-, counter- and bi-directionally pumped schemes, as well as dual-order pumped schemes.