A novel class of collocation methods based on the weighted integral form of ODEs

Abstract

In this work, a novel class of collocation methods for numerical integration of ODEs is presented. Methods are derived from the weighted integral form of ODEs by assuming that a polynomial function at individual time increment approximates the solution of the ODE. A distinct feature of the approach, which we demonstrated in this work, is that it allows the increase of accuracy of a method while retaining the number of method coefficients. This is achieved by applying different quadrature rule to the approximation function and the ODE, resulting in different behaviour of a method. Quadrature rules that we examined in this work are the Gauss–Legendre and Lobatto quadrature where several other quadrature rules could further be explored. The approach has also the potential for enhancing the accuracy of the established Runge–Kutta-type methods. We formulated the methods in the form of Butcher tables for convenient implementation. The performance of the new methods is investigated on some well-known stiff, oscillatory and non-linear ODEs from the literature.

Appendices

Appendix 1

In the appendix, tableaus for the presented families of methods are provided (for m = 2, 3, 4) in the form of analytical expressions. The following tables include:

• Tables 3, 4, and 5: tableaus for the Gm|Gm method;

• Tables 6, 7 and 8: tableaus for the Lm|Lm method;

• Tables 9, 10 and 11: tableaus for the Lm|Lm + 1 method;

• Tables 12, 13 and 14: tableaus for the Lm|Gm method.

Table 3 G2|G2 method

Table 4 G3|G3 method

Table 5 G4|G4 method

Table 6 L2|L2 method

Table 7 L3|L3 method

Table 8 L4|L4 method

Table 9 L2|L3 method

Table 10 L3|L4 method

Table 11 L4|L5 method

Table 12 L2|G2 method

Table 13 L3|G3 method

Table 14 L4|G4 method

Appendix 2

In the appendix, tableaus for the presented families of methods are provided (for m = 2, 3, 4) in the form of numerical values. The following tables include:

• Tables 15, 16 and 17: tableaus for the Gm|Gm method;

• Tables 18, 19 and 20: tableaus for the Lm|Lm method;

• Tables 21, 22 and 23: tableaus for the Lm|Lm + 1 method;

• Tables 24, 25 and 26: tableaus for the Lm|Gm method.

Table 15 G2|G2 method

Table 16 G3|G3 method

Table 17 G4|G4 method

Table 18 L2|L2 method

Table 19 L3|L3 method

Table 20 L4|L4 method

Table 21 L2|L3 method

Table 22 L3|L4 method

Table 23 L4|L5 method

Table 24 L2|G2 method

Table 25 L3|G3 method

Table 26 L4|G4 method