Euler algorithm, explicit implicit

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)... 

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... 

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