Euler scheme

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

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

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. 

The error will depend on the numerical approximation used for the SDEs. Here we use a simple Euler scheme to allow us to find analytical expressions for the errors. In addition to these equations, the behavior of the notional particles at the boundaries of the domain must also be specified. Here we avoid this difficulty by considering only notional particles far from the domain boundaries. 

Implicit schemes are unconditionally stable, this is shown in Fig. 8.27 where the evolution of the temperature, in a/ag steps, for values of a/ag higher than 0.5 is shown. Higher values of a/ag mean that we can use higher At, which at the end implies lower computational cost and faster solutions. The results in Fig 8.27 were obtained with the fully implicit Euler scheme, i.e., to = 1. The comparison between the implicit Euler and the Crank-Nicholson, to = 0.5 is illustrated in Fig. 8.28 for the center line temperature evolution. Although there is no apparent significance difference, we expect that the CN scheme is more accurate due to its second order nature. 

Figure 8.30 Steady-state temperature development using an implicit Euler scheme.

A. Nyberg and T. Schlick, ]. Chem. Phys., 95, 4986 (1991). A Computational Investigation of Dynamic Properties with the Implicit-Euler Scheme for Molecular Dynamics Simulation. 

In this example the property tp is advanced using the explicit Euler scheme for all the operators. However, both implicit and explicit methods can be employed. In order to take a larger time steps, implicit methods are generally preferred. The convective and diffusive terms can be further split into their components in the various coordinate directions, for example, by use of the Strang [181] operator factorization scheme. 

By use of the explicit Euler scheme, the discrete form of the governing equations can be written as [3] ... 

The convective terms were solved using a second order TVD scheme in space, and a first order explicit Euler scheme in time. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the smoothness monitor of van Leer [193] and the monotonic centered limiter [194]. The diffusive terms were discretized with a second order central difference scheme. The time-splitting scheme employed is of first order. 

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. 

The transient term is discretized using the implicit Euler scheme. 

Using an implicit Euler scheme to update the parameter vector (and the definition in Eq. (8.111) to rewrite the right-hand side), we find for grid cell i ... 

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 ... 

With standard DQMOM, when the explicit Euler scheme in time and the first-order upwind differencing scheme for space are employed, the volume-average weights in the cell centered at X at time (n + l)Af are... 

When DQMOM-FC is used with an explicit Euler scheme for time, the equations... 

Thus, the explicit Euler scheme with DQMOM-EC has a conservation error of order aC for moments of order greater than one when the weights and abscissas vary smoothly in time. Note that the coefficient for the error term has a form similar to the correction term in Eq. (D.5). Using the definitions of A and fi in Eqs. (D.38) and (D.39), the coefficient in Eq. (D.45) can be rewritten as... 

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... 

This corresponds to using the adjoint symplectic Euler scheme to solve the Newtonian part of the Langevin dynamics SDE, followed by an exact OU solve. An alternative is to use velocity Verlet for the Hamiltonian part, resulting in a scheme denoted BABO] with associated map... 

Talay, D. Stochastic Hamiltonian dissipative systems exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Relat. Fields 8, 163-198 (2002)... 

The integrals over the systematic parts of (3.73,74) can be evaluated by any number of standard quadrature methods. However, the simple Euler scheme will be sufficient to illustrate the essential result. Applying this method to (3.73,74) we obtain... 

Ccxnputation times also follow the expected evolution, increasing with the niunber of function evaluations per step required by the method. The Adams-Moulton scheme, however, which requires only two e uations per step, like the modified Euler scheme, was only slightly faster than the fourth-order Runge-Kutta due to the amount of computation involved in the predictor and corrector formulas. The former also provided extremely low errors, when applicable, but showed a tendency to become unstable at higher step sizes. 

Assuming a forward difference Euler scheme (Reddy, 1993)... 

The two constitutive laws presented in this work were obtained considering an isotropic material at a reference state, i.e., a state without any influence of external parameters such as temperature. The constitutive parameters evolutions are instead taking into account within the numerical simulation and thus the constitutive laws have to be discretized using an Euler scheme. A case study using the strain based internal state variables approach has been presented. This study shows how creep/relaxation could influence the results of an industrial problem such as the "baking" of a carbonaceous ramming paste. 

In order to model pattern formation in chemical systems at the macroscopic level, we must be able to solve the reaction-diffusion equation [1]. The simplest numerical method that one can use to solve this equation is an Euler scheme, where space is divided into a regular grid of points with separation Ax. In Figure 2, we show such a grid for a two-dimensional system. Each grid point is labeled by r = (f, ). Time is also divided into small segments At. 

Starting with the initial condition, Cg, we go to the eqitation and evaluate f(cg), the time derivative, or slope of the trajectory c(t). We then use this value to make a prediction of the state of the system at the next time step, Cj. We then repeat the process go to the eqitation with Cj to evaluate f(cj) and use the Euler scheme to predict and so on. 

Here, the Us(t) are the concentrations of all the macromolecules with length n at time t, represented by vector Us(t) Stot is the dimension (chain length) of the system, for polymer systems typically very large, up to 10 . For an approximation Ui of the solution u(t + r) after a time step from t to t + r a semi-(linear)-implidt Euler scheme is applied [Eq. (6), where = u(t), A is the derivative fu( >), and I the identity matrix]. 

To integrate Eqs. (2) and (3), we use the Euler scheme. Ax = 1.0, dt = 0.05. This discretized version of the TDGL equation is regarded as a kind of the cell-dynamic systems [10], which has been used in recent numerical studies on ordering processes in quenched systems. We performed computer simulation runs in two dimensions. 

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