Improved Euler method

Simulation by the improved Euler method has shown that a significant radiative heat transfer must be present before reaction zone migration can be demonstrated. 

Another improvement on the basic Euler method is to approximate the slope in the middle of the interval by the average of the slopes at the end points, or... 

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. 

In the improved Euler method (5.14) we use derivative information at two points of the time interval of interest, thereby increasing the order of the method. A straightforward extension of this idea is to use the derivative at several grid points, leading to the -step formulas... 

One problem with the Euler method is that it estimates the derivative only at the left end of the time interval between Z and. A more sensible approach would be to use the average derivative across this interval. This is the idea behind the improved Euler method. We first take a trial step across the interval, using the Euler method. This produces a trial value x + = jc, + the tilde above the... 

X indicates that this is a tentative step, used only as a probe. Now that we ve estimated the derivative on both ends of the interval, we average /(x ) and /(x +,), and use that to take the real step across the interval. Thus the improved Euler method is... 

This method is more accurate than the Euler method, in the sense that it tends to make a smaller error = x(f )-x for a given step size At. In both cases, the error —> 0 as Az —> 0, but the error decreases faster for the improved Euler method. One can show that E = Az for the Euler method, but E (Az) for the improved Euler method (Exercises 2.8.7 and 2.8.8). In the jargon of numerical analysis, the Euler method is first order, whereas the improved Euler method is second order. 

P 2.8.4 Redo Exercise 2.8.3, using the improved Euler method. 

Error estimate for the improved Euler method) Use the Taylor series arguments of Exercise 2.8.7 to show that the local error for the improved Euler method is G(Ar ). 

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. 

The Euler method can be improved by doing the calculation in two steps ... 

The resulting set of differential equations was solved numerically by means of an improved Euler method with variable step size control [9]. Parameter optimization was accomplished by a Mead Nelder simplex algorithm [10]. 

The midpoint method is an improvement on the Euler method in that it uses information about the function at a point other than the initial point of the interval. As seen in Figure 6.4, the values of x andy at the midpoint are used to calculate the step across the whole interval, h. 

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. 

The accuracy of the Euler method can be improved by utilizing a combination of forward and backward differences. Note that the first forward difference of y at i is equal to the first backward difference of y at (i + 1) ... 

In order to calculate the summation involved in Eq. (7.12), two methods were utilised. The first method involved directly calculating the summation, however the series converges very slowly when VcuID is large, making the method quite restrictive. The second method improves the convergence of Eq. (7.12) using the Euler transform, which takes a series of the form... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

There are iterative methods (e.g., Jacobi, Gauss-Seidel, Newton) whose purpose is simply to provide solutions for the steady-state equations, others (e.g., Euler and its improved versions) aim to give trajectories. Cycling will be felt as a disagreeable iteration artifact in the first case, as an indication of a probably cyclic trajectory in the second case. The relation between the behavior in a simple iteration method (e.g., Jacobi) and the real trajectory is interesting, if not simple. Consider, for instance, a simple negative loop comprising three inhibitory elements ... 

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