High-order Runge-Kutta Methods Research Articles

A high-order multi-time-step scheme for bond-based peridynamics

A high-order multi-time-step scheme (MTS) for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge–Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard STS scheme for PD by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.

Quantum-Inspired Neural Network with Runge-Kutta Method

In recent years, researchers have developed novel Quantum-Inspired Neural Network (QINN) frameworks for the Natural Language Processing (NLP) tasks, inspired by the theoretical investigations of quantum cognition. However, we have found that the training efficiency of QINNs is significantly lower than that of classical networks. We analyze the unitary transformation modules of existing QINNs based on the time displacement symmetry of quantum mechanics and discover that they are resembling a mathematical form similar to the first-order Euler method. The high truncation error associated with Euler method affects the training efficiency of QINNs. In order to enhance the training efficiency of QINNs, we generalize QINNs' unitary transformation modules to the Quantum-like high-order Runge-Kutta methods (QRKs). Moreover, we present the results of experiments on conversation emotion recognition and text classification tasks to validate the effectiveness of the proposed approach.

High order accuracy scheme for modeling the dynamics of predator and prey in heterogeneous environment

The aim of this work is to develop a compact finite-difference approach for modeling the dynamics of predator and prey based on reaction-diffusion-advection equations with variable coefficients. Methods. To discretize a spatially inhomogeneous problem with nonlinear terms of taxis and local interaction, the balance method is used. Species densities are determined on the main grid whereas fluxes are computed at the nodes of the staggered grid. Integration over time is carried out using the high-order Runge-Kutta method. Results. For the case of one-dimensional annular interval, the finite-difference scheme on the three-point stencil has been constructed that makes it possible to increase the order of accuracy compared to the standard second-order approximation scheme. The results of computational experiment are presented and comparison of schemes for stationary and non-stationary solutions is carried out. We conduct the calculation of accuracy order basing on the Aitken process for sequences of spatial grids. The calculated values of the effective order accuracy for the proposed scheme were greater than the standard two: for the diffusion problem, values of at least four were obtained. Decrease was obtained when directional migration was taken into account. This conclusion was also confirmed for non-stationary oscillatory regimes. Conclusion. The results demonstrate the effectiveness of the derived scheme for dynamics of predator and prey system in a heterogeneous environment.

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström extended phase space symplectic-like methods with the midpoint permutation.

Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid

Numerical calculation provides essential tools for deep space exploration, which are indispensable to mission design and planetary research. In a specific case of binary asteroid defense such as the DART mission, an accurate understanding of the possible dynamical responses and the system’s stability and engineers’ prerequisite. In this paper, we discuss the numeric techniques for tracking the year-long motion of the secondary after being perturbed, based upon which its rotational stability is analyzed. For long-term simulations, we compared the performances of several integrating schemes in the scenario of a post-impact full two-body system, including low- and high-order Runge–Kutta methods, and a symplectic integrator that combines the finite element method with a symplectic integral format. For rotational stability analysis of the secondary, we focus on the rotation of the secondary around its long-axis. We calculated a linearised error propagation matrix and found that, in the case of tidal locking of the secondary to the primary, the rotation is stable; as the perturbation amplitude of the spin angular velocity of the secondary increases, the rotation will lose stability and will be prone to being unstable in the long-axis direction of the secondary. Furthermore, we investigated the one-year-long influences of the non-spherical perturbations of the primary and the secondary.

Numerical higher-order Runge-Kutta methods in transient and damping analysis

Transient analysis of an RLC circuit (or LCR circuit) comprising of a resistor, an inductor, and a capacitor are analyzed. Kirchhoff’s voltage and current laws were used to generate equations for voltages and currents across the elements in an RLC circuit. From Kirchhoff’s law, the resulting first-order and second-order differential equations, The different higher-order Runge-Kutta methods are applied with MATLAB simulations to check how changes in resistance affect transient which is transitory bursts of energy induced upon power, data, or communication lines; characterized by extremely high voltages that drive tremendous amounts of current into an electrical circuit for a few millionths, up to a few thousandths, of a second, and are very sensitive as well important their critical and careful analysis is also very important. The Runge-Kutta 5th and Runge-Kutta 8th order methods are applied to get nearer exact solutions and the numerical results are presented to illustrate the robustness and competency of the different higher-order Runge-Kutta methods in terms of accuracy.

Convolutional neural networks combined with Runge–Kutta methods

A convolutional neural network can be constructed using numerical methods for solving dynamical systems, since the forward pass of the network can be regarded as a trajectory of a dynamical system. However, existing models based on numerical solvers cannot avoid the iterations of implicit methods, which makes the models inefficient at inference time. In this paper, we reinterpret the pre-activation Residual Networks (ResNets) and their variants from the dynamical systems view. We consider that the iterations of implicit Runge-Kutta methods are fused into the training of these models. Moreover, we propose a novel approach to constructing network models based on high-order Runge-Kutta methods in order to achieve higher efficiency. Our proposed models are referred to as the Runge-Kutta Convolutional Neural Networks (RKCNNs). The RKCNNs are evaluated on multiple benchmark datasets. The experimental results show that RKCNNs are vastly superior to other dynamical system network models: they achieve higher accuracy with much fewer resources. They also expand the family of network models based on numerical methods for dynamical systems.

Hybrid finite volume WENO and lattice Boltzmann method for shallow flows over erodible bed

A hybrid mesh-particle method is developed to simulate hydro- and morpho-dynamics in shallow flows. The well-balanced finite volume WENO and lattice Boltzmann methods were implemented to solve nonlinear shallow water and Exner equations, respectively and a splitting strategy coupled those two solvers. This strategy was known to suffer from numerical instability due to loss of hyperbolicity, but this problem was not observed in the present method. To achieve advanced interaction between two solvers, cross-computation and information exchange were performed at every step of the high-order Runge-Kutta method. Extensive numerical experiments verified whether the present hybrid method was implemented correctly, and the proposed method provided satisfactory results of the well-balanced property, the accuracy, and the stability in verification examples.

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach. Then the resulting system that satisfies the quadratic energy conservation law is discretized in time by combining explicit high-order Runge–Kutta methods with orthogonal projection techniques. The proposed schemes are shown to share the order of explicit Runge–Kutta method and thus can reach the desired high-order accuracy. Moreover, the methods are energy-preserving and explicit because the projection step can be solved explicitly. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other structure-preserving methods.

Аналіз можливостей вибору та адаптації алгоритмів чисельної реалізації диференціальних динамічних моделей

The paper analyzes possibilities of selection and adaptation of computational algorithms in the implementation of differential dynamic models, i.e. in solving ordinary differential equations. The limited resources of computer-integrated systems determine the requirements for the computing and control systems’ performance, which indicates the urgency of target selection or adaptation of numerical methods for solving equations of dynamics objects. The process of improving numerical methods has a long history and does not stop until now. The growing complexity of the studied dynamic objects has led to the development of implicit methods for dynamics’ numerical analysis, but research shows that the use of implicit methods is justified, if we assume the use of a significant step of integrating the source system. In addition, it is found that the available results on the formalization of power methods for solving algebraic equations in the integration step and their adaptation in computer use are still insufficient to address their use in the study of complex dynamic objects. The integration step upper limit indicates the inexpediency of using high-order Runge-Kutta methods for the purposes of modeling the dynamics of the studied systems in real time. Accordingly, with regard to quadrature methods, the problem of formalizing the construction is not solved in general. Thus, the task of selecting the optimal method can be formulated as follows: to determine the numerical method for the modeled object’s dynamics equations’ integration, for which the required speed of the control system can be achieved, and the error of solving the dynamics equations does not exceed the specified value. The analysis of the properties of different groups of numerical methods is carried out, which makes it possible to conclude that in choosing the best method the initial set of the required methods should be formed based on the single-step methods of the Runge-Kutta type and the quadrature methods no higher than the fourth order. In implementing stationary modes, the initial group of methods should also include the multi-step methods — explicit and the «forecast-correction» type

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favorably to the inverse Lax-Wendroff method in [5,6] where analysis and numerical experiments have previously verified the presence of order reduction [5,6]. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all nonnegative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.

Optimized low-dispersion and low-dissipation two-derivative Runge–Kutta method for wave equations

The paper is devoted to the optimization of the explicit two-derivative sixth-order Runge–Kutta method in order to obtain low dissipation and dispersion errors. The method is dependent on two free parameters, used for the optimization. The optimized method demonstrates the lowest dispersion error in comparison with other widely-used high-order Runge–Kutta methods for hyperbolic problems. The efficiency of the method is demonstrated on the solutions of problems for five linear and nonlinear partial differential equations by the method-of-lines. Spatial derivatives are discretized by finite differences and Petrov–Galerkin approximations. The work-precision and error—CPU time plots, as the dependence of CPU time on space grid resolution, are considered. Also, the optimized method compared with other two-derivative methods. In most examples, the optimized method has better properties, especially for the cases of computations with adaptive stepsize control at high accuracy. The lowest CPU time takes place for the optimized method, especially in the cases of fine space grids.

Equations that Describe Waves in Tubes with Elastic Walls and Analysis of Numerical Methods for Reversible and Weakly Dissipative Systems

Models of tube with elastic walls are presented: with controlled pressure, filled with incompressible fluid, filled with compressible gas. The non-linear theory of hyperelasticity is applied. The walls of a tube are described with complete membrane model. Equations are solved numerically. General approach for calculation of non-dissipative and weakly dissipative systems is given. Three-layer time and space centered reversible numerical scheme and similar two-layer space reversible numerical schemes with approximation of time derivatives by high-order Runge-Kutta methods are analysed. A method of correction of numerical schemes by inclusion of terms with high-order derivatives is developed. Additional terms with high-order derivatives my be included also in order to calculate non-dissipative and dissipative shocks at the same time.

Stability analysis of high order Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

In this paper, a new class of implicit stochastic Runge-Kutta (SRK) methods is constructed for numerically solving systems of index 1 stochastic differential-algebraic equations (SDAEs) with scalar multiplicative noise. By applying rooted tree theory analysis, the family of coefficients of the proposed methods of order 1.5 are calculated in the mean-square sense. In particular, we derive some four-stage stiffly accurate semi-implicit SRK methods for approximating index 1 SDAEs with scalar noise. For these methods, first, MS-stability functions, applied to a scalar linear test equation with multiplicative noise, are calculated. Then, their regions of MS-stability are compared with the corresponding MS-stability region of the original SDE. Accordingly, we illustrate that the proposed schemes seems to have good stability properties. Numerical results will be presented to check the convergence order and computational efficiency of the new methods.

Coupled transverse and torsional vibrations of the marine propeller shaft with multiple impact factors

An appropriate assessment of the dynamic behavior of a ship's marine propeller shaft is essential to enable the optional power delivery to the propeller and to minimize various vibrations during the rotation. As the mechanism of coupled transverse-torsional vibrations for the shaft is not fully revealed, which seriously affect the stability and reliability of ship's navigation. An accurate and applicable numerical model for the coupled vibrations of marine propeller shaft is thus proposed to solve this problem. This non-linear model with ordinary differential equations is analytical calculated on the basis of high order Runge-Kutta method. Experiments are conducted to validate the applicability of the proposed model through the comparison with numerical calculation, over a range of rotational speeds. The impact factors including eccentricity of cross section, damping coefficient and length-diameter ratio are discussed by comparing the Poincare surface of section. And the influence on the transient accelerations of the coupled vibrations are investigated in detail. The optimized design for marine propeller shaft is thus realized based on the adjustment of the unbalance quantity and structural dimension.

Optimization of high-order diagonally-implicit Runge–Kutta methods

This article presents constrained numerical optimization of high-order linearly and algebraically stable diagonally-implicit Runge–Kutta methods. After satisfying the desired order conditions, undetermined coefficients are optimized with respect to objective functions which consider accuracy, stability, and computational cost. Constraints are applied during the optimization to enforce stability properties, to ensure a well-conditioned method, and to limit the domain of the abscissa. Two promising third-order methods are derived using this approach, labelled SDIRK[3,(1,2,2)](3)L_14 and SDIRK[3,1](4)L_SA_5. Both optimized schemes have a good balance of properties. The relative error norm of the latter, the L2-norm scaled by a function of the number of implicit stages, is a factor of two smaller than comparable methods found in the literature. Variations on these methods are discussed relative to trade-offs in their accuracy and stability properties. A novel fifth-order scheme SDIRK[5,1](5)L_02 is derived with a significantly lower relative error norm than the comparable fifth-order A-stable reference method. In addition, the optimized scheme is L-stable. The accuracy and relative efficiency of the Runge–Kutta methods are verified through numerical simulation of van der Pol's equation, as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder, and in the turbulent wake of a NACA 0012 airfoil. These results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high-order Runge–Kutta methods with a balance between competing objectives.

Classical and Higher Order Runge-Kutta Methods with Nonlinear Shooting Technique for Solving Van der Pol (VdP) Equation

The goal of the research work is to examine the improvement of numerical solution of VdP equation. The well-known VdP equation is governed by the second order nonlinear ODE and then solved numerically using the classical Runge-Kutta (RK) method, RK-Fehlberg method of order five, Verner method of order eight and Cash-Karp method of order six with nonlinear shooting technique. In this work, numerical simulations have been carried out using NVdP code which is written in MATHEMATICA. Also, the accuracy and efficiency of the solution of VdP equation using different RK methods with nonlinear shooting technique has been investigated. For analysis of accuracy, the approximate exact solution obtained by perturbation method is used for the comparison. It is observed that all the different RK methods give accurate result of the VdP equation. But the classical RK method shows slightly better performance than the other single step techniques.
 Dhaka Univ. J. Sci. 66(1): 43-47, 2018 (January)

Lagrangian analysis of high-speed turbulent premixed reacting flows: Thermochemical trajectories in hydrogen–air flames

A Lagrangian analysis approach is used to examine the effects of high-speed turbulence on thermochemical trajectories in unconfined, stoichiometric hydrogen–air (H2–air) premixed flames. Two different intensities of turbulence in the unburnt reactants are considered, giving premixed flames with Karlovitz numbers of roughly 150 and 450. These two cases are modeled using direct numerical simulations (DNS) with both multi- and single-step H2–air reaction kinetics. In each of the four resulting simulations, trajectories of fluid parcels are calculated using a high-order Runge–Kutta method, and time series of temperature and chemical composition within each parcel are recorded. The resulting thermochemical trajectories are used to examine the evolution of thermodynamic quantities and chemical composition, as well as measure fluid parcel residence times and path lengths during different phases of the combustion process. Fuel mass fraction and temperature within fluid parcels are shown to be frequently non-monotonic along fluid trajectories in both single- and multi-step H2–air simulations, and the prevalence of non-monotonic trajectories increases with increasing turbulence intensity. Using results from single-step simulations, it is shown that this non-monotonicity can be caused solely by molecular transport processes resulting from large gradients in temperature and species concentrations created by turbulent advection. As a related consequence of advection, fluid parcel residence times are found to be smaller than in a laminar flame and the ratio of turbulent to laminar residence times decreases from roughly 0.8 to 0.6 as the turbulence intensity increases. By contrast, fluid parcel path lengths in the present high-speed turbulent flames are found to be substantially greater than laminar path lengths, resulting in fluid parcels that travel 4 and 7 times further than in a laminar flame for the two different turbulence intensities considered here.

High order Runge–Kutta methods for impulsive delay differential equations

The purpose of this paper is to obtain high order Runge–Kutta methods for nonlinear impulsive delay differential equations (IDDEs). In order to achieve the purpose, continuity and smoothness of the exact solutions of IDDEs are analyzed. And then the discontinuous points and non-smooth points are chosen as a part of the nodes of the Runge–Kutta methods. Furthermore, it is proved that the pth order Runge–Kutta method for IDDEs is also convergent of order p. And some simple numerical examples are given to demonstrate the theoretical results.