Euler stabilization

Crisco 11, Panjabi MM, Yamamoto I, Oxland TR (1992) Euler stability of the human ligamentous lumbar spine. Part II Experiment, Clinical Biomechanics 7 27-32... 

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... 

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]. 

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. 

The Cauchy problem for a system of differential equations of first order. Stability condition for Euler s scheme. We illustrate those ideas with concern of the Cauchy problem for the system of differential equations of first order... 

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. 

Second method consists of a straightforward discretization method first order (Euler) explicit in time and finite differences in space. Both the time step and the grid size are kept constant and satisfying the Courant Friedrichs Lewy (CFL) condition to ensure the stability of the calculations. To deal with the transport part we have considered the minmod slope limiting method based on the first order upwind flux and the higher order Richtmyer scheme (see, e.g. Quarteroni and Valli, 1994, Chapter 14). We call this method SlopeLimit. 

The stability of the Euler method is improved by using interpolation instead of extrapolation, and considering the tangent evaluated at 4+1 ... 

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 explain the stability characteristics of the forward Euler algorithm, consider the following model problem [215] ... 

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 explicit Euler method stability criterion was given as... 

Evaluate the stability criteria as above for a stepsize of h = 10-6 s for the three equations. Is the explicit Euler method predicted to be stable for this stepsize hi... 

Because the central-differencing on the continuity equation is only neutrally stable, it is beneficial to introduce an artificial damping term to assist numerical stability [199]. Following common practice in solving Euler equations, a first-order damping term may be... 

Proof. Each face orbit has either a trivial stabilizer or is fixed by a rotation of order 2, 3, or 5. If some face / of gonality m has a stabilizer of order j = 2,3,5, then we have m = kj j, and the orbit of / contains exactly y faces. Together with Euler formula (1.1), this implies ... 

Finally Brubaker and Euler conclude from a conformational analysis that the stability difference between -cw- 3 and A-ci s-(3-oxalato-N,N -bis(2-picolyl)l,2-(S)-diaminopropanecobalt(III) (A-cis-fi- [Co(pnpic)C204 ]+) must be at least 3 kcal/mole58. ... 

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