Euler stabilization method

CLQA = corrected local quadratic approximation DDRP = dynamically defined reaction path DRP = dynamic reaction path ES = Euler stabilization method GS = Gonzalez and Schlegel method IMK = Ishida-Morokuma-Kormomicld method LQA = local quadratic approximation MB = Miillar-Brown method MEP = minimum energy path ODE = ordinary differential equations SDRP = steepest descent reaction path VRl = valley-ridge inflection. 

The method of Ishida, Morokuma, and Komomicki (IMK) is shown in Figure 3(a). It is a modification of the explicit Euler method that adds a stabilization step hence it is also known as the Euler stabilization method (ES). An explicit Euler step of length a is taken from along the tangent... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

Flow in a Slit. Turning to a slit geometry, a flat velocity profile gives the simplest possible solution using Euler s method. The stability limit is independent of y ... 

Example 8.8 Explore conservation of mass, stability, and instability when the convective diffusion equation is solved using the method of lines combined with Euler s method. 

Another way to study the stability of explicit equations is to use the positivity theorem. For Euler s method, the equations can be written in the form... 

Numerical solutions to PDEs must be tested for convergence as Ar and Az both approach zero. The flnite-difference approximations for radial derivatives converge O(Ar ) and those for the axial derivative used in Euler s method converge 0(Az). In principle, just keep decreasing Ar and Az until results with the desired accuracy are achieved, but it turns out that Ar and Az cannot be chosen independently when using the method of lines. Instead, values for Ar and Az are linked through a stability requirement that the overall coefficient on the central dependent variable cannot be negative ... 

We aim at the development of fully robust, stable methods and therefore we restrict our attention to implicit methods with Particularly, we shall consider two cases with 6 = Yi and 0 = 1, which correspond to the Crank - Nicolson (CN) and backward Euler (BE) method, respectively. More details can be found e.g. in Quarteroni Valli (1994). Finally, we chose the BE method for its higher stability (the CN scheme can show some local oscillations for large time steps). 

The fourth-order explicit Runge-Kutta algorithm has a slightly better stability region than the Euler forward method. 

What is more relevant for using Verlet to simulate molecular dynamics is the remarkable stability of energy shown in the right-hand panel of Fig. 2.4. Notice that the energy error does not appear to exhibit a steady accumulation in time (unlike for Euler s method, where it exhibited a linear-in-time growth). The explanation for this unexpected behavior lies in the structural properties of the method, a topic we explore in this and the next chapter. 

This defines the stability region for Euler s method as a disk in the complex hX-plane centered around —1 of radius 1. This region is shown in Fig. 4.1. If hX, where A is an eigenvalue of A lies in the indicated stability region, then Euler s method will be stable for the linear system z = Az. 

The eigenvalues are ico. The reason a method like Euler s method can never perform well for molecular dynamics is that molecular dynamics is a Hamiltonian system and at the bottoms of basins on the energy surface, which correspond to stable centers, we expect all the eigenvalues of a local linearization of the problem to be purely imaginary. The stability condition always fails to hold and, for Euler s method, z grows exponentially rapidly away from the equilibrium point. 

Fig. 4.1 Stability regions for Euler s method (l ) and Symplectic Euler/Verlet (right). When a harmonic oscillator is treated using these methods, the origin is unstable for Euler s method, regardless of stepsize—this means that there is no choice of scaling h which will allow us to ensure that 11 + ftA, < 1. On the other hand, the Verlet method has an interval of stability on the imaginary axis, and it is always possible to find a value of h which guarantees that hQ < 2...

Applying a Runge-Kutta method to such a linear system allows direct determination of the stability condition. For example, Euler s method would yield... 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

A high degree of accuracy is not called for in many calculations of the evolution of environmental properties because the mathematical description of the environment by a reasonably small number of equations involves an approximation quite independent of any approximation in the equations solution. Figure 2-3 shows how the accuracy of the reverse Euler method degrades as the time step is increased, but it also shows the stability of the method. Even a time step of 40 years, nearly five times larger than the residence time of 8.64 years, yields a solution that behaves like the true solution. In contrast, Figure 2-2 shows the instability of the direct Euler method a time step as small as 10 years introduces oscillations that are not a property of the true solution. 

The key feature of the systems to be considered in this book is that they have short memories that is, the effects of perturbations diminish with the passage of time. In the example of this chapter, the carbon dioxide pressure returns to a value of 1 within a century or two of the perturbation, regardless of the size of the initial perturbation. In this kind of system, computational errors do not grow as the calculation proceeds instead, the system forgets old errors. That is why the reverse Euler method is useful despite its simplicity and limited accuracy. The many properties of the environment that are reasonably stable and predictable can, in principle, be described by equations with just this kind of stability, and these are the properties that can be simulated using the computational methods described in this book. 

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