Implicit Euler

Ft has been shown [1] that the Euler implicit formula is more stable than the explicit one. The stability of these methods will be discussed in Sec. 3.7. 

The implicit-Euler (IE) scheme, for example, discretizes system (1) as ... 

T. Schlick, S. Figueroa, and M. Mezei. A molecular dynamics simulation of a water droplet by the implicit-Euler/Langevin scheme. J. Chem. P%s., 94 2118-2129, 1991. 

G. Zhang and T. Schlick. The Langevin/implicit-Euler/Normal-Mode scheme (LIN) for molecular dynamics at large time steps. J. Chem. Phys., 101 4995-5012, 1994. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

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... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In the implicit Euler scheme, the imknown function at the new time step appears on the right-hand side. 

A method which represents a compromise between the explicit and implicit Euler... 

The Crank-Nicolson method is popular as a time-step scheme for CFD problems, as it is stable and computationally less expensive than the implicit Euler scheme. 

Three more parameters are implicitly included which are the Euler angles that describe the orientation of the principal axes system. 

The integration of the state equations (Equation 10.21) by the fully implicit Euler s method is based on the iterative determination of x(t1+i). Thus, having x(t,) we solve the following difference equation for x(t, i). 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

Thus the implicit methods become slower and slower as the number of ODEs increases, despite the fact that large step sizes can be taken. Therefore plain old explicit Euler turns out to run faster than the impheit methods on many reahstically large problems, unless the stiffness of the system is very, very severe. We will talk more about this in Chap. 5. 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

Once all of the conditions were determined and parameters chosen, the equations were solved by an implicit Euler method. The program was written with a self adjusting step size and analytic Jacobian to reduce error and run time. 

The truncation errors in (5.9) and (5.12) are of the same magnitude, but the implicit Euler method (5.11) is stable at any positive step size h. This conclusion is rather general, and the implicit methods have improved stability properties for a large class of differential equations. The price we have to pay for stability is the need for solving a set of generally nonlinear algebraic equations in each step. 

To compare the explicit and implicit Euler methods we exploited that the solution (5.3) of (5.2) is known. We can, however, estimate the truncation error without such artificial information. Considering the truncated Taylor series of the solution, for the explicit Euler method (5.7) we have... 

The formulas (5.7) and (5.11) of explixit and implicit Euler methods, respectively, are unsymmetrical, using derivative information only at one end of the time interval of interest. Averaging the slopes of the two tangent lines means using more information, and gives... 

As stated above, the spatial derivative is approximated without regard to the time level. The distinction between explicit and implicit solutions depends on the time level at which the spatial derivatives are evaluated. Finite-difference stencils for explicit and implicit Euler methods are illustrated in Fig. 4.13. 

Fig. 4.13 Finite-difference stencils for the explicit and implicit Euler methods. The spatial index is j and the time index is n. For equally spaced radial mesh intervals of dr, rj = (j — 1 )dr, 1 < j < J. For equally spaced time intervals, tn = (n — 1 )dt, n > 1.

Implicit methods, which have far better stability properties than explicit methods, provide the computational approach to solving stiff problems. The simplest implicit method is the backward (implicit) Euler method, which is stated as... 

Compared to the explicit Euler method (Eq. 15.9), note that the right-hand side is evaluated at the advanced time level tn+1- If f(t, ) is nonlinear then Eq. 15.22 must be solved iteratively to determine yn+. Despite this complication, the benefit of the implicit method lies in its excellent stability properties. The lower panel of Fig. 15.2 illustrates a graphical construction of the method. Note that the slope of the straight line between y +i and yn is tangent to the nearby solution at tn+, whereas in the explicit method (center panel) the slope is tangent to the nearby solution at t . 

The first three terms represent the implicit Euler algorithm and the remaining [bracketed] term represents the local truncation error. A Taylor series expansion about tn+ (in the negative t direction) yields an expression for y(t )... 

As discussed in Section 15.3.2 on the implicit solution of transient differential equations, one step of the backward Euler method takes the form... 

The implicit Euler integration method is examined here. We will use the set of two sequential reactions, rate constants, and initial conditions described in the previous problem. Note This problem uses results from tasks 1-3 in the previous problem.)... 

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