Explicit integration algorithms

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

In ab initio calculations //, is calculated from Eq. 5.3 by actually performing the integration using explicit mathematical expressions for the basis functions , and 4>j and the Hamiltonian operator H of course the integration is done by a computer following a detailed algorithm. How this algorithm works will now be outlined. 

To illustrate the difference in stability properties between explicit and implicit integration algorithms, consider again the equation used to describe valve dynamics in Section 2.2. Dropping the subscripts from equation (2.9) for clarity and generality, and setting the demanded valve travel, xj, to zero, indicating a... 

The velocities do not explicitly appear in the Verlet integration algorithm. The velocities can be calculated in a variety of ways a simple approach is to divide the difference in positions at times t + St and t — St by 2St ... 

The explicit expression for defined in equation (9.21) may be derived by requiring that for all k the vanishes at all times. Using a certain numerical integration algorithm, the set of Lagrangian multipliers is no longer evaluated by equation (9.21) but by equations that contain free parameters corresponding to the. These... 

The finite element discretization requires solving the dynamic equation of motion (Eq. 1) by means of an explicit time marching integration algorithm. In this type of discretization, mass can be distributed over layer thickness. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

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... 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

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