Runge-Kutta variable step size integration

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. 

The results shown in Figs. 3.1 and 3.2 were obtained with the scheme of Eq. (3.24) with At = 0.01 s and a Runge-Kutta integration of second order with variable step size (cf. [9]). The difference between the two procedures may be neglected in this case. It is obvious that the thermal explosion (runaway) leaves but little time for emergency interventions. 

Equations (9.1), (9.2), and (9.3) are ordinary differential equations in which distance is the independent variable. The technique of integration is to start from a perturbed full equilibrium condition at the hot boundary of the ffame and integrate backwards across the ffame by an explicit method. Dixon-Lewis et al. (1979a,b 1981) used a fourth-order Runge-Kutta procedure with variable step size for this purpose. We continue here by reviewing brieffy the application of the method with both partial equilibrium and quasi-steady-state assumptions. 

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... 

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