Embedded Pair of Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations (original) (raw)
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We constructed a new fifth order five-stage singly diagonally implicit Runge-Kutta (DIRK) method which is specially designed for the integrations of linear ordinary differential equations (LODEs). The restriction to linear ordinary differential equations (ODEs) reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. The best strategy for practical purposes would be to choose the coefficients of the Runge-Kutta methods such that the error norm is minimized. Thus, here the error norm obtained from the error equations of the sixth order method is minimized so that the free parameters chosen are obtained from the minimized error norm. The stability aspect of the method is also looked into and found to have substantial region of stability, thus it is stable. Then a set of test problems are used to validate the method. Numerical results show that the new method is more efficient in terms of accuracy compared to the existing method.
Mathematical Problems in Engineering, 2015
We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the formy(iv)=f(x,y)denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them.
A method for solving ordinary differential equations has been developed using implicit Runge-Kutta methods. The implicit Runge-Kutta methods used are based in two quadratures of Lobatto type. The first quadrature produces the principal Runge-kutta method which is of sixth order, while the second quadrature produces a Runge-Kutta method of third order which is embedded in the former. Both implicit Runge-Kutta methods constitute the Lobatto embedding form of third and sixth orders with four stages. The most important advantage of this method is that it is implicit only in the second and third stages, which reduces considerably the costs of computer calculations. The Butcher notation is used here for the analysis of the studied methods. In order to solve, for each step, the system of non-linear equations in the implicit auxiliary variables k_2 and k_3, an explicit Runge-Kutta method of four stages and fourth order is defined for the same intermediate points, such as the implicit sixth order Runge-Kutta method. This explicit method estimates the inicial values for the aforementioned auxiliary variables, and then, an iterative method of the type ``fixed point'' is used to solve the system of non-linear equations for each step. With the third order Runge-Kutta method, an estimation of the local truncation error may be calculated using a comparison with the sixth order method. This aspect is used to control the step size when tolerances for the relative and absolute global errors are specified. An algorithm is presented to do this step control automatically. The implicit method, as is exposed here, is really useful and has demonstrated to be efficient to solve huge and stiff systems of ordinary differential equations. Finally, convergence criteria and stability analysis are studied for the implicit Runge-Kutta methods presented here.
MODIFIED DIAGONALLY IMPLICIT RUNGE-KUTTA METHODS
The experimental evidence indicates that the implementation of Newton's method in the numerical solution of systems of ordinary differential equations (ODE's) y' =[(t, y), y(a)= Yo, [a, b] by implicit computational schemes may cause difficulties. This is especially true if (i) [(t, y) and/or ['(t, y) are quickly varying in and/or y and (ii) a low degree of accuracy is required. Such difficulties may also arise when diagonally implicit Runge-Kutta methods (DIRKM's) are used in the situation described by (i) and (ii). In this paper some modified DIRKM's (MDIRKM's) are derived. The use of MDIRKM's is an attempt to improve the performance of Newton's method in the case where [ and ['y are quickly varying only in t. The stability properties of the MDIRKM's are studied. An error estimation technique for the new methods is proposed. Some numerical examples are presented.
2019
Three Diagonally Implicit Two Derivative Runge-Kutta (DITDRK) methods for the numerical solution of first order Initial Value Problems (IVPs) are derived. We present fourth, fifth and sixth-order Diagonally Implicit Two Derivative Runge-Kutta methods designed with minimum number of function evaluations. The stability of the method derived are analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods of the same order and properties.
A FIFTH ORDER FIVE STAGE (5(5)) DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING IVPs OF ODEs
Bukunmi Ebenezer Afolabi, 2021
Ordinary Differential Equations are useful tools for describing real life phenomenon. However, several ordinary differential equations cannot be solved either by analytical approach or semi-analytic approach. This study is concerned with the numerical integration of first order ordinary diffrential equations. Here in, to which we develop a fifth order five-stage diagonally implicit Runge-Kutta (DIRK) method. The fifth order five stage DIRK methods have region of absolute stability within the interval [-4.5, 0] and [-0.85, 0]. The method also shows favourable result on implementation on some first order ODE.
A FOURTH ORDER FOUR STAGE (4(4)) DIAGONALLY IMPLICIT RUNGE-KUTTA METHOD FOR SOLVING IVPs OF ODEs
Bukunmi Ebenezer Afolabi, 2021
Numerical integration for ordinary differential equations have been of great interest since some ordinary differential equations arising from real life phenomenon cannot be integrated by analytical procedures. The interest of this study is to develop a new fourth order four-stage Diagonally Implicit Runge-Kutta (DIRK) method for the selection of IVPs of ODEs. The 4(4) DIRKM were Constructed by using the Taylor series expansion and implemented on some test problems. Our proposed method suggests that they perform favourably and are suitable for IVPs of ODEs.