Implicit Runge-Kutta Method Of Order Research Articles

Derivation of Five-Stage Implicit Runge-Kutta Method of Order 10 via Gauss-Legendre Quadrature for Ordinary Differential Equations

This paper presents developing and implementing a five-stage implicit Runge-Kutta method of order ten via Gauss-Legendre quadrature for stiff or oscillatory first-order initial value problems (IVPs) of ordinary differential equations (ODEs). Using continuous collocation and interpolation techniques, the implicit Runge-Kutta method was developed by combining Legendre polynomials and exponential functions as the basis function. The properties of the method were investigated, and it was shown that it is consistent and A-stable. The new method was evaluated on two sampled problems involving stiffness and oscillation. The numerical results demonstrate that the new implicit Runge-Kutta is computationally efficient and outperforms previous methods of similar derivations.

Explicit K-Symplectic and Symplectic-like Methods for Charged Particle System in General Magnetic Field

We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter ε in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods.

Optimal error estimates of the local discontinuous Galerkin methods based on generalized fluxes for 1D linear fifth order partial differential equations

In this paper, we study the error estimates of local discontinuous Galerkin methods based on the generalized numerical fluxes for the one-dimensional linear fifth order partial differential equations. We use a newly developed global Gauss–Radau projection to obtain the linear type of optimal error estimates. The numerical experiments show that the scheme coupled with the third order implicit Runge–Kutta method can achieve the optimal (k+1)th order of accuracy.

A coupled neutronic/thermal–hydraulic module for the transient analysis of VVER-1000 reactor during reactivity insertion accidents

For the safety analysis of nuclear power plants, it is essential to provide a simple and fast model to investigate the time-dependent behavior of reactor cores. Most research papers in this field have generally been limited to use coupling techniques between existing codes. In this paper, an integrated module called CMS (nuclear reactor Core Modular Simulator) is proposed to simulate the transient thermo-neutronic behavior of a VVER-1000 reactor core during reactivity insertion accidents. CMS contains three modules: a neutronic module based on the point kinetic equations, a thermal-hydraulic module based on the time-dependent single heated channel approximation and a reactivity calculation one. A two-stage fourth order implicit Runge-Kutta method was used for solving the point kinetics equations and the homogeneous two-phase flow model was considered for the thermal-hydraulic part. To verify the proposed module, two reactivity insertion accidents of the Bushehr VVER-1000 reactor, a design basis accident and an anticipated operational occurrence, have been simulated and the obtained results have been compared with its final safety analysis report showing a good agreement.

The Scheme of 10th Order Implicit Runge-Kutta Method to Solve the First Order of Initial Value Problems

Abstract—To construct a scheme of implicit Runge-Kutta methods, there are a number of coefficients that must be determined and satisfying consistency properties and Butcher’s simplifying assumptions. In this paper we provide the numerical simulation technique to obtain a scheme of 10th order Implicit Runge-Kutta (IRK10) method. For simulation process, we construct an algorithm to compute all the coefficients involved in the IRK10 scheme. The algorithm is implemented in a language programming (Turbo Pascal) to obtain all the required coefficients in the scheme. To show that our scheme works correctly, we use the scheme to solve Hénon-Heiles system.Keywords—ODEs, 10th order IRK method, numerical technique, Hénon-Heiles system

English

In this paper we develop a more efficient three-stage implicit Runge-Kutta method of order 6 for solving first order initial value problems of ordinary differential equations. Collocation method is used to derive Continuous schemes in which both the interpolation and collocation points are at perturbed Gaussian points. This gives a higher order scheme, which is more efficient and stable than the existing similar ones. Simple linear problems are used to check its level of accuracy and stability. Key words: Implicit, more efficient, stable, collocation methods, Perturbed Gaussian points and error estimates.

Two-dimensional cascadic finite element computations of combustion problems

We present an integrated time-space adaptive finite element method for solving systems of two-dimensional nonlinear parabolic systems in complex geometry. The partial differential system is first discretized in time using a singly linearly implicit Runge-Kutta method of order three. Local time errors for the step size control are defined by an embedding strategy. These errors are used to propose a new time step by a PI controller algorithm. A multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space. The local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation. This two-fold adaptivity successfully ensures an a priori prescribed tolerance of the solution. The devised method is applied to laminar gaseous combustion and to solid-solid alloying reactions. We demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm.

One-step implicit methods or solving delay differential equations

Two one-step implicit methods—the second order Trapezium method and the fourth order implicit Runge-Kutta method for solving the delay differential equations (DDE) are developed. The significance of implicit methods lie in their 4-stability for ordinary differential equations. Different techniques are used to approximate the delay term. We also discuss the local truncation error estimate. Numerical examples are solved to show the effectiveness of the methods so developed.