In this paper, a third order General Linear Method for finding the numerical solution of Volterra integro-differential equation is considered. The order conditions of the proposed method are derived based on techniques of B-series and 'rooted trees'. The integral operator in Volterra integro-differential equation approximated using Simpson's rule and Lagrange interpolation is discussed. To illustrate the efficiency of third order General Linear Method, we compare the method with a third order Runge-Kutta method.
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Table 1. Matrix representation of coefficients of GLM.
Table 2.
Matrix coefficients of GLM with
Table 3. Order conditions of GLM up to order three.
No | Order conditions |
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 | |
Table 4. Coefficients Set 1 of third order GLM
Table 5. Coefficients Set 2 of third order GLM
Table 6. Coefficients Set 3 of third order GLM
Table 7. Maximum global errors for Problem 1
GLM, |
RK, |
|
Step size | MAXE | |
Table 8. Maximum global errors for Problem 2
GLM, |
RK, |
|
Step size | MAXE | |
Table 9. Maximum global errors for Problem 3
GLM, |
RK, |
|
Step size | MAXE | |
Table 10. Maximum global errors for Problem 4
GLM, |
RK, |
|
Step size | MAXE | |
Table 11. Maximum global errors for Problem 5
GLM, |
RK, |
|
Step size | MAXE | |
Table 12. Total number of function evaluations Problems 1 - 5
GLM, |
RK, |
|
Step size | TFE | |
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