Explicit Runge-Kutta Method Of Order Research Articles

Consistency and Convergence Analysis of an 𝐹(𝑥,𝑦) Functionally Derived Explicit Fifth-Stage Fourth-Order Runge-Kutta Method

The purpose of this paper is to analyze the consistency and convergence of an explicit fifth-stage fourth-order Runge-Kutta method derived using 𝒇(𝒙,𝒚) functional derivatives. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in a previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

Method of Lines for the Chiral Nonlinear Schrödinger Equation

In this paper, we solve chiral nonlinear Schrodinger equation (CNSE) numerically. Two numerical methods are derived using the explicit Runge-Kutta method of order four and the linear multistep method (Predictor-Corrector method of fourth order). The resulting schemes of fourth order accuracy in spatial and temporal directions. The CNSE is non-integrable and has two kinds of soliton solutions: bright and dark soliton. The exact solutions and the conserved quantities of CNSE are used to display the efficiency and robustness of the numerical methods we derived. Interaction of two bright solitons for different parameters is also displayed.

Strong Stability Preserving Properties of Composition Runge–Kutta Schemes

In this paper strong stability preserving (SSP) properties of Runge–Kutta methods obtained by composing k different schemes with different step sizes are studied. The SSP coefficient of the composition method is obtained and an upper bound on this coefficient is given. Some examples are shown. In particular, it is proven that the optimal $$n^2$$ -stage third order explicit Runge–Kutta methods obtained by Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) are composition of first order SSP schemes.

A Splitting Method for the Benjamin-Bona-Mahoney Equation Using WENO Scheme

The Benjamin-Bona-Mahoney equation can be split into a system of an elliptic equation and an ordinary differential equation (ODE). For the elliptic equation, we use a classical finite difference weighted essentially non-oscillatory (WENO) scheme. For the ODE, the third order explicit Runge-Kutta method is employed to discretize the time derivative. Due to the WENO reconstruction, the splitting method shows an excellent ability in capturing the formation and propagation of shock peak on solutions. The numerical simulations for asymptotic solution of the BBM equation illustrate the capability of the method.

Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations

Some effects of surface tension on fully-nonlinear, long, surface water waves are studied by numerical means. The differences between various solitary waves and their interactions in subcritical and supercritical surface tension regimes are presented. Analytical expressions for new peaked travelling wave solutions are presented in the case of critical surface tension. The numerical experiments were performed using a high-accurate finite element method based on smooth cubic splines and the four-stage, classical, explicit Runge-Kutta method of order four.

Slip Analysis at Fluid-Solid Interface in MHD Squeezing Flow of Casson Fluid through Porous Medium

An unsteady squeezing flow of Casson fluid having Magneto Hydro Dynamic effect and passing through porous medium channel with slip at the boundaries has been modelled and analyzed. Similarity transformations are applied to the governing partial differential equations of the Casson model to get a highly non-linear fourth order ordinary differential equation. The obtained equation is then solved analytically using the Homotopy Perturbation Method (HPM) for uniform and non-uniform slip at the boundaries. Five cases of boundary conditions, representing slip at upper wall only, uniform slip at both walls, non-uniform slip where slip at upper wall is greater than that of lower wall, non-uniform slip where slip at lower wall is greater than that of upper wall, and slip at lower wall only are considered and thoroughly investigated. Validation is performed by solving the equation numerically using fourth order explicit Runge Kutta method (ERK4). Both analytical and numerical results show good agreement. Lastly, the effects of various fluid parameters on the velocity profile are investigated for each case graphically. Analysis of these plots show that the positive and negative squeeze numbers have opposite effect on the velocity profile throughout all the cases. It is also observed that various fluid parameters like Casson, MHD, and Permeability have similar effects on the velocity profile in the cases when slip is occurring at the upper wall only, and non-uniform slip at both the boundaries with slip at lower wall is greater than upper wall. Furthermore, similar effects have been observed when slip is uniform at both the boundaries, and in case of non-uniform slip with slip at lower wall is less than the upper wall.

Stability Analysis of the Inverse Lax–Wendroff Boundary Treatment for High Order Central Difference Schemes for Diffusion Equations

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem are treated by the simplified inverse Lax–Wendroff procedure. For the fully discrete case, a third order explicit Runge–Kutta method is used as an example for the analysis. Stability is analyzed by both the Gustafsson, Kreiss and Sundstrom theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

Unsteady Squeezing Flow of Casson Fluid with Magnetohydrodynamic Effect and Passing through Porous Medium

An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel is modeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of non-Newtonian fluid to a highly nonlinear fourth-order ordinary differential equation (ODE). The obtained boundary value problem is solved analytically by Homotopy Perturbation Method (HPM) and numerically by explicit Runge-Kutta method of order 4. For validity purpose, we compare the analytical and numerical results which show excellent agreement. Furthermore, comprehensive graphical analysis has been made to investigate the effects of various fluid parameters on the velocity profile. Analysis shows that positive and negative squeeze numberSqhave opposite effect on the velocity profile. It is also observed that Casson parameterβshows opposite effect on the velocity profile in case of positive and negative squeeze numberSq. MHD parameterMgand permeability constantMphave similar effects on the velocity profile in case of positive and negative squeeze numbers. It is also seen that, in case of positive squeeze number, similar velocity profiles have been obtained forβ,Mg, andMp. Besides this, analysis of skin friction coefficient has also been presented. It is observed that squeeze number, MHD parameter, and permeability parameter have direct relationship while Casson parameter has inverse relationship with skin friction coefficient.

Multi-scale Runge-Kutta_Galerkin method for solving one-dimensional KdV and Burgers equations

In this paper, the multi-scale Runge-Kutta_Galerkin method is developed for solving the evolution equations, with the spatial variables of the equations being discretized by the multi-scale Galerkin method based on the multi-scale orthogonal bases in Hm0(a,b) and then the classical fourth order explicit Runge-Kutta method being applied to solve the resulting initial problem of the ordinary differential equations for the coefficients of the approximate solution. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection-diffusion problem), the KdV equation (single solitary and 2-solitary wave problems) and the KdV-Burgers equation, where analytical solutions are available for estimating the errors. Numerical results show that using the algorithm we can solve these equations stably without the need for extra stabilization processes and obtain accurate solutions that agree very well with the corresponding exact solutions in all cases.

Modeling and analysis of unsteady axisymmetric squeezing fluid flow through porous medium channel with slip boundary.

The aim of this article is to model and analyze an unsteady axisymmetric flow of non-conducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.

On the Component Analysis and Transformation of An Explicit Fifth – Stage Fourth – Order Runge – Kutta Methods

This work is designed to transform the fifth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This is because the rooted trees and derived equations for the y derivatives and x,y derivatives are the same for the explicit fourth-stage fourth-order methods, hence, we are motivated to analyze the fifth-stage fourth-order method. This idea is also derivable from general graphs and combinatorics.

Analysis of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries Using OHAM

In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is studied with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations is reduced to a single fourth order ordinary differential equation. The resulting boundary value problems are solved by optimal homotopy asymptotic method (OHAM) and fourth order explicit Runge-Kutta method (RK4). It is observed that the results obtained from OHAM are in good agreement with numerical results by means of residuals. Furthermore, the effects of various dimensionless parameters on the velocity profiles are investigated graphically.

Strong-stability-preserving, Hermite-Birkhoff time-discretization based on k step methods and 8-stage explicit Runge-Kutta methods of order 5 and 4

Ruuth and Spiteri have shown, in 2002, that fifth-order strong-stability-preserving (SSP) explicit Runge-Kutta (RK) methods with nonnegative coefficients do not exist. One of the purposes of the pr...

A multiscale Runge-Kutta Galerkin method for one-dimensional sine-Gordon equations

In this paper the multiscale Runge-Kutta Galerkin method is presented for solving sine-Gordon equations. The multiscale Galerkin method based on the multiscale orthonormal bases constructed in [8] is used to discrete the spacial variable, and the classical four order explicit RungeKutta method is applied to solve the resulting nonlinear ordinary differential equations. Because of the strong expression of the multiscale bases and the stability of the Runge-Kutta method, stable and accurate approximate solutions are obtained in relatively low dimensional subspaces. All numerical results illustrate the effectiveness of the proposed algorithm. Mathematics Subject Classification: 65J15; 65M60

Strong-stability-preserving, Hermite–Birkhoff time-discretization based on [formula omitted] step methods and 8-stage explicit Runge–Kutta methods of order 5 and 4

Ruuth and Spiteri have shown, in 2002, that fifth-order strong-stability-preserving (SSP) explicit Runge–Kutta (RK) methods with nonnegative coefficients do not exist. One of the purposes of the present paper is to show that the Ruuth–Spiteri barrier can be broken by adding backsteps to RK methods. New optimal, 8-stage, explicit, SSP, Hermite–Birkhoff (HB) time discretizations of order p, p=5,6,…,12, with nonnegative coefficients are constructed by combining linear k-step methods of order (p−4) with an 8-stage explicit RK method of order 5 (RK(8, 5)). These new SSP HB methods preserve the monotonicity property of the solution and prevent error growth; therefore, they are suitable for solving hyperbolic partial differential equations (PDEs) by the method of lines. Moreover, these new HB methods have larger effective SSP coefficients and larger maximum effective CFL numbers than Huang’s hybrid methods and RK methods of the same order when applied to the inviscid Burgers equation. Generally, HB methods combined with RK(8, 5) have maximum stepsize 24% larger than HB combined with RK(8, 4).

Free surface flow around a ship model using an interface-capturing method

A free surface, turbulent flow code around ship hulls using the Slightly Compressible Flow formulation is used for simulating the flow around a Wigley hull. The code uses finite differences and the numerical scheme was marched in time using the 2nd order explicit Runge–Kutta method. Turbulent flows were simulated using the Baldwin–Lomax turbulence model. The grid is generated using an algebraic grid generator. The free surface treatment method is an interface-capturing one, on a fixed grid on which the kinematic free surface equation is solved. Several boundary conditions were implemented and their behavior was assessed. The flow conditions are the ones for which experimental and numerical results exist. The results show very good agreement with the experimental results.

Numerical treatment of Kap's equation using a class of fourth order method

Kap’s equation is a stiff initial value problem. This paper deals with the treatment of Kap’s equation using a class of 4th order explicit Runge–Kutta method. Numerical computation was carried out using Microsoft Visual C++. The results of the computation were found to be highly accurate and consistent with minima errors. A comparison of the results generated from the scheme was also carried out vis-a-vis some other conventional explicit Runge–Kutta formulae. The proposed class of method was found to compare favourably well. Key words: Kap’s equation, stiffness, initial value problems, multiderivative explicit Runge-Kutta method, fourth-order method.

An assessment of the boundary conditions for free surface simulations using an interface-capturing method

A free surface, turbulent flow code around ship hulls using the Slightly Compressible Flow formulation is used for simulating the flow around a Wigley hull. The code uses finite differences and the numerical scheme was marched in time using the 2nd order explicit Runge-Kutta method. Turbulent flows were simulated using the Baldwin-Lomax turbulence model. The grid is generated using an algebraic grid generator. The free surface treatment method is an interface-capturing one, on a fixed grid on which the kinematic free surface equation is solved. Several boundary conditions were implemented and their behavior was assessed. The flow conditions are the ones for which experimental and numerical results exist. The results show good agreement with the experimental results.

On the 25 stage 12th order explicit Runge-Kutta method

On the 25 stage 12th order explicit Runge-Kutta method

An experimental study on iterative methods to compute transient solutions of large Markov models

In this article, we report results of an experimental study on six iterative methods to compute the transient probabilities of large Markov models: full matrix exponentiation, forward Euler method, explicit Runge-Kutta methods of order 2 and 4 and Adams-Bashforth multi-steps methods of order 2 and 4. We suggest a simple but efficient implementation of these algorithms. We discuss how to tune their few parameters. We present experimental results that contradict the literature.