By Thomas K. McInerny, Henry M. Adam, Deborah E. Campbell, American Academy of Pediatrics
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11) are an example of an implicit Runge Kutta method (again, a one-step method, since the solution at i + 1 requires only the solution at i). We can also consider multistep or multivalue methods for initial value ODEs that require more than one past 3 A 2 × 2 (two linear constant coefﬁcient ODEs in two unknowns) will have two eigenvalues, λ = a + ib 1 1 1 and λ2 = a2 + ib2 . If these two eigenvalues are real (b1 = b2 = 0), the analytical solution will consist of two real exponentials, ea1 t , ea2 t which produce a stable solution if a1 < 0, a2 < 0.
Thus, in the case of eqs. 3), a 51 point grid in z produced 51 ODEs with 51 eigenvalues. These eigenvalues were not computed as part of the MOL analysis, but for our discussion, all we require is an understanding that when the eigenvalues are widely separated with respect to their real parts (a stiff system), many small steps are required to compute a complete solution. 34 Introduction to PDE integration in space and time constant coefﬁcient ODEs. The reason why this combination presents a computational problem is the requirement of a small h (for stability) and a large problem time scale so that many small steps are required to compute a complete solution.
Clearly the additional derivative evaluations for the calculation of k1 to k4 were worthwhile. In general, this is a valid 1 The machine epsilon, eps, is the smallest number for which the computer will consider 1 + eps greater than 1. In other words, it is a measure of the precision of the computer arithmetic. In the case of Matlab, eps ≈ 10−15 so that the calculations can be performed with a maximum precision of about 1 part in 1015 . , 32-bits or 64-bits per word are common formats. The value of the machine epsilon can be controlled by the analyst to some extent, for example, by using double or quadruple precision arithmetic.