Euler method stability

The explicit Euler method stability criterion was given as... 

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. 

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. 

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 . 

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... 

This set of ordinary 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 or AA. If Eulers method is used for integration, the time step is limited by... 

The standard Euler methods and Runge-Kutta methods do not converge for stiff ODE S. A still system can be defined as one in which the stability of the numerical methods used becomes an issue. Maple has an inbuilt stiff solver. 

The formula is known both as the improved Euler formula and as Heun s method [170]. This method has roughly the stability of the explicit Euler method. Unfortunately, the stability may not be improved by iterating the corrector because this iteration procedure converges to the trapezoid rule solution only if At is small enough. 

There are three major problems for the Euler method. First, the accuracy is poor, since the method is based upon Eq. (F.16), in which only a first-order difference expression is used. The errors in the method are proportional to At. Second, stability is difficult to achieve for many problems. The only way to have a stable Euler method is to use a small enough time step-size, but you may not know what value is sufficient. Furthermore, a value that is sufficient at the beginning may not be sufficient later on, and it may take an excessively long time to finish the computation. Third, to validate the results it is necessary to solve the problem at least twice, with different time-steps. The method can, however, be programmed in Excel, as Figures A.3 and A.4 demonstrate. 

This is a second-order Runge-Kutta method (Finlayson, 1980), sometimes called the midpoint rule. The first step is an approximation of the solution halfway between the beginning and ending time, and the second step evaluates the right-hand side at that mid-point. The error goes as (At), which is much smaller than that achieved with the Euler method. The second-order Runge-Kutta methods (there are several) also have a stability limitation. 

Figure 2.4 Stability region of the forward Euler method, linear equation to do so ...

This is the so-called linear stability condition of the Symplectic Euler method if hQ < 2 the integrator is stable. When hQ > 2, the eigenvalues of the discretization method are both real, with one strictly inside and one strictly outside the unit circle. This implies that the method will exhibit exponentially growing solutions. We say that the stability threshold of the Symplectic Euler method is 2/f2. 

In terms of stability, the explicit Euler method is unstable if the step size is not properly chosen. The implicit methods, such as the backward Euler and the trapezoidal methods, are stable, but the solution may oscillate if the step size is not chosen small enough. The illustrative example in the next section reveals the source of the stability problem. 

Numerical integration of a problem usually gives rise to results that are unusual in the sense that often the computed values blow up. The best example of this so-called stability problem is illustrated in the numerical integration of dy/dt = -y using the Euler method with a veiy large step size. 

The step size for the explicit Euler method must be smaller than 2/A to ensure stability. 

If the explicit Euler method is used, the value of p is 2 (see section 7.3 on stability for more details), and hence the maximum step size to maintain numerical stability is... 

All the terms in the right-hand side of Eq. 12.137 are known, and hence, this makes the forward difference in time rather attractive. However, as in the Euler method (Chapter 7), this forward difference scheme in time suffers the same handicap, that is, it is unstable if the grid size is not properly chosen. Using the stability analysis (von Rosenberg 1969), the criterion for stability (see Problem 12.12) is... 

To resolve the problem of stability, we approach the problem in the same way we did in the backward Euler method. We evaluate Eq. 12.130 at the unknown time level fy+j and use the following backward difference formula for the time derivative term (which is first order correct)... 

The explicit (Euler) method described above has this stability limitation. There are other methods that do not. One of them (reverting again to the ode 21) is the backward difference (or backward implicit, BI) formula ... 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

If this IS the case, then after multiple steps the error will decrease and eventually disappear. Under these conditions the Euler Method for this problem is stable. Rewriting this singuality leads to the following condition for absolute stability... 

If this were the only context in which CML models were used, their utility would be severely limited. For values y beyond the stability limit, the Euler method fails and one obtains solutions that fail to represent the solutions of the reaction-diffusion equation. However, it is precisely the rich pattern formation observed in CML models beyond the stability limit that has attracted researchers to study these models in great detail. Coupled map models show spatiotemporal intermittency, chaos, clustering, and a wide range of pattern formation processes." Many of these complicated phenomena can be studied in detail using CML models because of their simplicity and, if there are generic aspects to the phenomena, for example, certain scaling properties, then these could be carried over to real systems in other parameter regimes. The CML models have been used to study chemical pattern formation in bistable, excitable, and oscillatory media." ... 

Figure 4.3 Stability behavior of the explicit Euler method when applied to the unconstrained truck with different step sizes...

In Fig. 4.8 the stability region of the three stage Radau Ila method is displayed. One realizes that the stability of the Radau method is much alike the stability of the implicit Euler method, though the three stage Radau method has order 5. Again we note the property i (0) = 1 which corresponds to zero stability in the multistep case. 

In order to give an introduction to the most important concepts used in different methods, such as accuracy, stability, and efficiency, we will start with the exphcit Euler method, which is the simplest numerical method to use when solving ODEs. At this point, we should stress that the accuracy of this method is low, and that it is only conditionally stable. For this reason, it is not used in practice to solve any engineering problems. However, it serves well to illustrate a method for solving ODEs numerically, and for introducing the concepts of accuracy and stability. Later in this chapter more advanced numerical methods will be presented methods that have higher accuracy and better stability properties, and that are implemented in modem software products. 

All the methods presented so far, e.g. the Euler and the Runge-Kutta methods, are examples of explicit methods, as the numerical solution aty +i has an explicit formula. Explicit methods, however, have problems with stability, and there are certain stability constraints that prevent the explicit methods from taking very large time steps. Stability analysis can be used to show that the explicit Euler method is conditionally stable, i.e. the step size has to be chosen sufficiently small to ensure stability. This conditional stability, i.e. the existence of a critical step size beyond which numerical instabilities manifest, is typical for all explicit methods. In contrast, the implicit methods have much better stability properties. Let us introduce the implicit backward Euler method. 

In contrast to the forward (explicit) Euler method, which uses the slope at the left-hand side to step across the interval, the imphcit verison of the Euler method crosses the interval by using the slope at the right-hand side, as shown in Figure 6.8. The implicit formula does not give any direct approximation ofy +i, instead an iterative method, e.g. the Newton method, is added inside the loop, thus advancing the differential equation to solve fory +1. This obviously comes at the price of more computation, but allows stability... 

Finally the solution converges at z = 0.916079783099616, which is the reactant concentration, y, predicted by the implicit Euler method at the first time step i = 0.1. This procedure is repeated for the following nine time steps to determine the final reactant concentration. To sum up, the implicit Euler method involves more computation it does not improve accuracy because it is only first-order accurate, but it significantly improves stability. 

Methods similar to this are commonly used to solve the momentum equation when an accurate time history of the flow is required. In these particular cases, more accurate time advancement methods than the first order Euler method must be used. Explicit methods are preferred for (fast) transient flows because they are more accurate and less computationally demanding than implicit methods with the same time step size. To allow longer time steps to be used and for stability reasons, some of the terms may also be treated implicitly. 

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... 

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