Ordinary differential equations Runge-Kutta methods

But87] Butcher J. C. (1987) The Numerical Analysis of Ordinary Differential Equations Runge-Kutta and General Linear Methods. Wiley, Chichester. 

The system of Eqs. (7.4.34)-(7.4.42) was put in the form of ordinary differential equations using the method of characteristics (e.g., [43]) and the finite difference form of the resultant expressions was integrated numerically using the Runge-Kutta method. Some computed results are shown in Figs. 7.21-7.23 together with actual experimental measurements. 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

Ordinary differential equations the Runge-Kutta method... 

Figure 3.6 Numerical solution of ordinary differential equations sketch of the four steps of the Runge-Kutta method to the order four giving the n+1 th estimate y(n+1) from the nth estimate y ". ...

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

For the simulation of the reactor behaviour the system of ordinary differential equations was integrated by means of a Runge-Kutta-Merson method with variable step length, whereas the nonlinear algebraic equations were solved by a Newton-Raphson iteration. 

In this chapter we will consider only ordinary differential equations, that is, equations involving only derivatives of a single independent variable. As well, we will discuss only initial-value problems — differential equations in which information about the system is known at f = 0. Two approaches are common Euler s method and the Runge-Kutta (RK) methods. 

Equations 10.106 to 10.111 constitute a set of algebraic equations and first order ordinary differential equations. The two algebraic Eqs. 10.104 and 10.105 are solved using the classical procedure described by Villadsen [65]. The set of ordinary differential equations is easy to solve with the fourth-order Runge-Kutta method. 

Runge-Kutta-Gill method is the most widely used single-step method for solving ordinary differential equations. 

Runge-Kutta Fourth Order Method for a System of Ordinary Differential Equations... 

Numerical methods are required to integrate these coupled ordinary differential equations and to calculate the time-dependent molar density of each component in the exit stream of the first CSTR. Generic integral expressions are illustrated below. The Runge-Kutta-Gill fourth-order correct algorithm is useful to perform this task. 

And finally, the new values of the concentrations at the time r + Ar can be calculated using a numerical solution of this latter ordinary differential equation. In this case, Euler s method (Perry, Green, and Malone, 1984) is used due to its simplicity, although errors are proportional to AL Other method of high order, as Runge-Kutta (Perry, Green, and Malone, 1984), can be used if needed ... 

Runge-Kutta is one of the most popular methods for solving nonstiff ordinary differential equations because it is a self-starting method, that is, it needs only a condition at one point to start the integration, which is in contrast to the Adams family methods where values of dependent variables at several values of time are needed before they can be used. For the following equation... 

Many software packages have incorporated adaptive difference schemes for ordinary differential equations as the core of their designs. NDSolve of Mathematica and ode45 of MATLAB are two applications that are based on an adaptive extension of the fourth-order Runge-Kutta method. 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

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