Euler integration scheme

Use of the explicit Euler integration scheme will be illustrated by integrating the following set of sequential reactions... 

Schlick and Olson recently developed such an algorithm that permits larger time steps. The implicit Euler integration scheme is combined with the Langevin dynamics formulation, which contains frictional and random... 

In the case of an Euler integration scheme, approximate formulae were derived to estimate an appropriate and stable time interval. 

The first-order Euler integration scheme reduces the Langevin equation to... 

A. EULER ALGORITHM. The simplest possible numerical-integration scheme (and the most useful) is Euler (pronounced oiler ), illustrated in Fig. 4.7. Assume we wish to solve the ODE... 

This is the simplest possible numerical integration scheme. It is known as Euler s method. 

If a single-stage Euler explicit time-integration scheme is used, the updated moment set can be written as... 

Eq. (B.l). Thus, as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form ... 

For the spatial solution of the nonlinear coupled multi-field problem given in Sect 1.3, the Finite Element Method (FEM) is applied. The equations for the three fields are solved with a Newton-Raphson algorithm, and the time integration is performed with the implicit Euler backwards scheme. 

Figure 13 Steady-state energy of an ensemble of anions and cations in a 150-mM solution of KCl as a function of time step, for both the Euler and the Verlet-like integration schemes.

It should be noted that the numerical integration in Eq. (7.223) is similar to the backward-Euler numerical integration scheme. Therefore, the integrator in Fig. 7.154(c), with the output signal sampled when 0 is high, is known as the backward-Euler discrete integrator (BEDI). 

Forward-Euler discrete integrator (FEDI) A discrete-time integrator based on the forward-Euler numerical integration scheme. 

The Euler characteristic, %, of a closed surface is related to the local Gaussian curvature K r) via the Gauss-Bonnet theorem [Eq. (8)]. A number of different schemes have been proposed to calculate the local curvatures and the integral in Eq. (8). 

The PES in the vicinity of IRC is approximated by an (N - -dimensional parabolic valley, whose parameters are determined by using the gradient method. Specific numerical schemes taking into account p previous steps to determine the (p + l)th step render the Euler method stable and allow one to optimize the integration step in Eq. (8.5) [Schmidt et al., 1985]. When the IRC is found, the changes of transverse normal vibration frequencies along this reaction path are represented as... 

The volume fraction variables were assumed to be time dependent in the semi-implicit discretization scheme used to solve the continuity equations. Considering the last two terms in the above relation, the transient term was then approximated by an explicit Euler time discretization scheme followed by volume integration in which the transient term was kept constant. If we thereafter multiply the resulting relation by Atjp, the relation can be rewritten as ... 

The continuity equation is re-written on the integral form, integrated in time and over a grid cell volume in the non-staggered grid for the scalar variables sketched in Fig C.2. The transient terms are discretized with the implicit Euler scheme. 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

Fig. 7.5. Rotation of a twisted spiral. Left Spatial distribution of Rcij. Right Position of the spiral at times t = 0 (solid), ( = 340 (dashed), t = 680 (dotted). Parameters are a = 4.19, / = 0.992, v = 3.9895, B = 0.045.5, the system siz.e is 500 x. 500. Numerical integration using the explicit Euler scheme with Ax = 0.2 and At = 0.0025.

For the one-dimensional problem, integrated with a forward Euler scheme, the amplification of errors was found to be... 

The interest is usually in conserving some more complicated first integrals, for example the energy of the system. None of the schemes we have considered so far (Euler s method, Stormer-Verlet, etc.) conserves this quantity exactly, even in the... 

A simple first order example of a Langevin dynamics integrator is the method obtained by composing one of the Symplectic Euler variants with an Ornstein-Uhlenbeck step. To get a feel for how the expansion goes, let us work out its terms for such a splitting scheme in the case of a one degree of freedom model with unit mass H = + f/(g). To be explicit, let us say our numerical method first solves... 

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