Error Orders

The use of data of 4He and 3He vapour pressure which was accurately reported for T > 0.5 K was recommended. Unfortunately it was clear that also for the IPTS-68 errors (order of 10-4 K) existed in this temperature scale in comparison with the thermodynamic temperature. 

With hmax = ma,x(xi+i — x/), the global error order of the classical Runge-Kutta method is of order 4, or 0(h/nax), provided that the solution function y of (1.13) is 5 times continuously differentiable. The global error order of a numerical integrator measures the maximal error committed in all approximations of the true solution y(xi) in the computed y values y. Thus if we use a constant step of size h = 10 3 for example and the classical Runge-Kutta method for an IVP that has a sufficiently often differentiable solution y, then our global error satisfies... 

From the above treatment, the error orders of the approximations can be determined. First, a definition of what is meant here is required. With equal intervals of length h, orders are expressed as powers of that length. Here we have arbitrarily spaced points, and thus a set of different hk- In computations to confirm error order expectations, the following scheme can serve. Refer all hk as displacements from point i, as above (3.45). A given derivative can then be computed. Then, all points around the reference point Xj are moved to a given fraction a of their original displacements from the reference point, so that now there is a new set of displacements,... 

It turns out that the order of this approximation is also the global error order of the calculation using the Euler method. An alternative way to proceed is to go from the Taylor expansion for y(t + St), as in (3.3),... 

This formula has a global error of 0(6t2) and will here be called RK2. We can do even better, generating more fc s and getting higher orders. All these RK formulae, including RK2, have variants that have the same error orders. For example, RK2 can also be carried out by generating k2 as... 

This method seems at first sight unpromising, because of its low error order, the same as that for Euler. However, it has some very useful stability properties (see later) and forms the basis for several high-order methods, as will be seen. 

Extrapolation is an old technique invented by Richardson in 1927 [469]. Generally it makes use of known error orders to increase accuracy. In the present context, its application is based on the first-order method BI, mentioned above. One defines a notation in terms of operations L on the variable y(t), the operation being that of taking a step forward in time. Thus, the notation L y t) or, in terms of discrete time steps where one whole interval is St, Liyn, means a single step of one interval (the 1 being indicated by the subscript on L). The simplest variant is then the application of operation L, followed by two operations, t jA, that is, two consecutive steps of half St (again starting the first from yn), and finally a linear combination of the two results ... 

In publications providing Rosenbrock coefficients such as [100,347], there appear alternative coefficients, hatted , such as etc. These always provide another variant with an order smaller by one than the one used. The purpose of this is that the difference between the two forms provides an (over)estimate of the error. The practice is not followed in this book, as we are generally mainly interested in the error order. So these alternative, lower-order coefficients are not included in Appendix A. 

If we do not know an exact solution, we can still estimate the error order by Method 2, as described by Osterby [430], We must use one more interval size, a2/ . We then have a third result ... 

Table 14.1. Error orders (and errors at t 1) for some BDF starts, for the ode...

Molecular area or dimension, standard error, order parameter, shielding... 

For equally spaced points, the first derivative of function f(x) is approximated (to error order of Ajc ) as follows ... 

An increase in the order of the algorithm, p, or of its local error order m does not necessarily lead to an increase of accuracy. 

The root of both of these failures is that, within the discretization error, it is not possible to decide whether the particle moved from ti(tj) to r,(t,+i) outside the domain directly from element n or through element + 1. The assignment of a new position to the lost particle is therefore arbitrary, but sinee the number of such failures has always a higher error order, it is irrelevant where the position of such particles should be reinitialized, apart from obviously poor choices like... 

First of all, these 0(6T ) errors refer to a single 6T step, whereas in the simulation in the T-range 0 < T < 1 we take 6T steps. Generally, this reduces the error order by one. Tests show that it is in fact the discretisation of the second derivative which limits the accuracy, hence the 0(H ) error for all methods. We might say that, in view of this, we are lucky that the better methods (CN, RKI) give better results, since they suffer from the same error source. 

Error orders are given in brackets (measured by taking 100 steps)... 

The 3-point BDF KW start is actually very simple to implement, requiring only a 2 X 2 system whose solution (for yi and j2) is easily expressed, and so it could be feasible for use in pde. However, the table shows that it results in no better errors than simpH- or the rational start, so it does not recommend itself. It is interesting to note, regarding the error orders, that both simpH- and rational show an order close to 2, regardless of the BDF order, meaning that with these starts, BDF using more than three points is no improvement over three-point BDF. The only start that enables the full accuracy of higher BDF orders is the KW start, which follows the BDF order. 

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