Runge-Kutta methods integration step

The fourth order Runge-Kutta method is the workhorse for the numerical integration of ODEs. Elaborate routines with automatic step-size control are available in Matlab. 

With hmax = ma,x(xi+i — x/), the global error order of the classical Runge-Kutta method is of order 4, or 0(h/nax), provided that the solution function y of (1.13) is 5 times continuously differentiable. The global error order of a numerical integrator measures the maximal error committed in all approximations of the true solution y(xi) in the computed y values y. Thus if we use a constant step of size h = 10 3 for example and the classical Runge-Kutta method for an IVP that has a sufficiently often differentiable solution y, then our global error satisfies... 

Starting with an initial value of cA and given c,(t), Eq. (8-4) can be solved for cA(t + At). Once cA(t + At) is known, the solution process can be repeated to calculate cA(t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. As discussed in Sec. 3, more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations ... 

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. 

First, without explaining the details [15], we will develop an Excel spreadsheet for the numerical integration of the reaction mechanism 2as seen in Figure 7.13. The fourth-order Runge-Kutta method requires four evaluations of concentrations and derivatives per step. This appears to be a serious disadvantage, but as it turns out, significantly larger step sizes can be taken for the same accuracy, and the overall computation times are much shorter. We will comment on the choice of appropriate step sizes after this description. 

These results were obtained by numerically integrating the system with a Runge-Kutta method. The time step was 0.25, and 15,000 points were computed. Grassberger and Procaccia also report that the convergence was rapid the correlation dimension could be estimated to within 5 percent using only a few thousand points. ... 

The basic idea of the Runge-Kutta methods is illustrated through a simple second order method that consists of two steps. The integration method is constructed by making an explicit Euler-like trail step to the midpoint of the time interval, and then using the values of t and tp at the midpoint to make the real step across the whole time interval ... 

EROS handles concurrent reactions with a kinetic modeling approach, where the fastest reaction has the highest probability to occur in a mixture. The data for the kinetic model are derived from relative or sometimes absolute reaction rate constants. Rates of different reaction paths are obtained by evaluation mechanisms included in the rule base that lead to partial differential equations for the reaction rate. Three methods are available that cover the integration of the differential equations the GEAR algorithm, the Runge-Kutta method, and the Runge-Kutta-Merson method [120,121], The estimation of a reaction rate is not always possible. In this case, probabilities for the different reaction pathways are calculated based on probabilities for individual reaction steps. 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

To assure sufficient accuracy of the numerical integration, we repeatedly halved the integration step size until no significant difference (0.1% ) in solutions occurred. For the fourth-order Runge-Kutta method this required a step size of. 04 sec (43 steps). An upper bound of the total error introduced by the numerical integration procedure can be obtained for the Runge-Kutta method (29). At a step size of 0.04 sec, the error estimate calculated is 0.004% (14). 

For the integration of the differential equations a number of numerical standard methods exist, such as Runge-Kutta or multi step methods [31]. In the... 

A /(-order Runge-Kutta method requires at least p calculations of the system f(y, t) at each integration step the same-order explicit multistep method needs a single system evaluation. 

The formulae are predisposed for an integration with constant step. If we wish to change the integration step, several tough problems arise (e.g., a Runge-Kutta method has to be used as per the initialization problem). 

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