Euler algorithm, explicit

The explicit first-order Euler algorithm is used. The variables that we are solving for as functions of time are V and H. The right-hand sides of Eqs. (5.3) and (5.4) are the derivative functions. These are called VDOT and HDOT in the program. At the nth step in time... 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

Let us now apply Euler s explicit and implicit algorithms with a constant step size h. They are defined by the relationships (in vector notation)... 

S is nothing other than the stiffness ratio of the system. The number of steps for Euler s explicit algorithm increases with the stiffness of the system. This conclusion is quite general or, in other words, explicit algorithms are not suitable for integrating stiff systems. 

A similar expression holds for the values of v at each grid point, that is, for v j. We need to discretize the derivatives in time, as well, and this can be done in either an implicit or an explicit fashion. Considering first, for simplicity, an explicit Euler algorithm for the discretized time derivatives, the following system of equations will result ... 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

Sometimes the ODEs that arise in studies in nonlinear dynamics can be solved using explicit methods (such as the forward Euler) which require less computations per step and are thus cheaper and ter to implement. The Runge-Kutta femily of algorithms are a popular implementation of the explicit methods. Runge—Kutta methods begin with a Taylor series expansion the order of the particular Runge-Kutta method used is simply the highest order term retained in the Taylor series. 

The fourth-order explicit Runge-Kutta algorithm has a slightly better stability region than the Euler forward method. 

These methods (Euler, midpoint, and Runge-Kutta) are examples of explicit algorithms. For the same step size, the higher-order Runge-Kutta method gives better precision but at the cost of more calculations in each step. 

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