Runge-Kutta accuracy

A popular fourth-order Runge-Kutta method is the Runge-Kutta-Feldberg formulas (Ref. Ill), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

The 4th 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 an acceptable accuracy and overall the computation times are very much shorter. 

One important note for this system we had to increase the default accuracy of the integration (RelTol and AbsTol) and also use the stiff solver odel5s. We leave it to the reader to experience the Runge-Kutta solver ode45 or the default accuracy. 

Numerical simulations of (53) were performed using a central finite difference approximation of the spatial derivatives with fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. The typical system size was Lx = Ly = 256. Some test runs were made with Lx = Ly = 512 and 1,024. We used a uniform mesh size Sx Sy 1 and time step 8t = 2 x 10 2. The accuracy of the calculations was checked by choosing 8x = 8y = 0.5 and 8/ 2 x 10-3. The... 

The integration, between YA = Y. and YA = Y, can be carried out by the Runge-Kutta-Merson method. For gooa accuracy, the intervals have to be very small. For this calculation, the correct values of ky-k% must be known. If periodic deviations are obtained between the experimental (from the FCCU) yields and the yields predicted by this model, a corrective factor in these k values can be introduced. 

This method often requires very small integration step sizes to obtain a desired level of accuracy. Runge-Kutta integration has a higher level of accuracy than Euler. It is also an explicit integration technique, since the state values at the next time step are only a function of the previous time step. Implicit methods have state variable values that are a function of both the beginning and end of the current... 

Dimensionless parameter estimation The model s equations were solved using a fourth-order Runge-Kutta method. The dimensionless parameters estimation (Table I) was made as follows Only two of the dimensionless numbers Ai, Af, A2, A3, Ai and A5 are known directly. The parameter Ai can be estimated with reasonable accuracy since the catalyst surface area was measured independently ... 

After casting the equations in dimensionless form, they were solved by an Euler s method integration on the University of Minnesota Cyber 7000 computer system. The accuracy of this method was checked by a fourth order Runge-Kutta integration, which gave agreement to within 0.5%. 

Coefficients au and b, are determined in order that the algorithm possesses some qualities such as stability, accuracy, etc. A classical explicit fourth-order Runge—Kutta algorithm is defined by the values... 

The method is one way to handle a stiff set of odes, and is an extension of fourth-order explicit Runge-Kutta. The function to be solved is approximated over the next time interval by a combination of a linear function of the dependent variable and a quadratic function of time (assuming that it is strongly time-dependent) and this increases the accuracy and stability of the fourth-order Runge-Kutta method considerably. Today, however, we have other methods of dealing with stiff sets of odes, so this method might be said to have outlived its usefulness. 

Runge-Kutta Methods. The Runge-Kutta methods attempt to improve the accuracy of the calculation and avoid the evaluation of higher-order derivatives. The general approach is to determine values of the first-order differential equation at subintervals of the... 

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