Section 9: Use of Numerical Integration OptionsThe example Fig. 5.1 in Toride et al. (1995) is used to illustrate the use of numerical integration options. The parameters are input in D5:D13, and E11:G12. The time instants are put in B21:B221. The dimensionless pulse is calculated in cell G8 by formula =D9*D8/D5. The Peclet number is calculated in cell G9 by formula =D9*D5/D10.
The five numerical integration schemes are listed in N1:N5, and are used to create a list in cell L5 using menu Data->Validation. Similarly, FALSE and TRUE are listed in O1:O2, and are used to create a list in cell L6. The tolerance is input in L7 for the adaptive numerical integration schemes (Romberg, Simpson, and Lobatto), and the number of integration points are in L8 (Chebyshev and Gauss scheme). Then define strNI, LogTransform, tol, and nPoint for the L5, L6, L7, and L8. The formula for D21 is shown in Fig. 10.1, and can be extended to the rest of the cells for forward solution. Users can select different numerical integration schemes, including if time coordinates are log transformed, change the tolerance or number of integration points and compare the efficiency and accuracy of different schemes.
Fig. 9.1 Illustration of using different numerical integration scheme