Recently, I pay attention to the numerical integration, and I find an simple approach to calculate the result. The engineering problem is to calculate the volume of detention pond after the runoff hydrograph has acquired. In Section 6.1 of "Handbook of Detention Facilities Planning", the volume of detention pond is calculated by the hydrograph method. Actually, the so-called hydrograph method is the numerical integration for the intersection area of two hydrographic curves.
I find the polynomial equations of the curves by the techniques of curve fitting, and then I calculate the numerical integration by the Simpson's rule. However, there exists a simple approach to calculate the area, and it is to calculate the area by many rectangle areas. If you use very small △x, you can get the result which is almost equal to the result of Simpson's rule. It's very easy to do this, and you can obtain the result by Microsoft EXCEL, for example.
Why? If you still remember Calculus, you shall review the limit concepts. The rectangle method has the same limit with the Simpson's rule, so that you can get very close result between the two methods. Actually, there are three methods which are Simpson's rule, Trapezoidal rule and Rectangle method. However, compared with the Simpson's rule, the Trapezoidal rule or Rectangle method is NOT efficient for obtaining an accurate result. Consequently, you have to calculate the result by the three methods, and then you can decide a good △x for these methods respectively. It's quite fun.
References
I find the polynomial equations of the curves by the techniques of curve fitting, and then I calculate the numerical integration by the Simpson's rule. However, there exists a simple approach to calculate the area, and it is to calculate the area by many rectangle areas. If you use very small △x, you can get the result which is almost equal to the result of Simpson's rule. It's very easy to do this, and you can obtain the result by Microsoft EXCEL, for example.
Why? If you still remember Calculus, you shall review the limit concepts. The rectangle method has the same limit with the Simpson's rule, so that you can get very close result between the two methods. Actually, there are three methods which are Simpson's rule, Trapezoidal rule and Rectangle method. However, compared with the Simpson's rule, the Trapezoidal rule or Rectangle method is NOT efficient for obtaining an accurate result. Consequently, you have to calculate the result by the three methods, and then you can decide a good △x for these methods respectively. It's quite fun.
References
- Water Resources Planning Institute, Water Resources Agency, Ministry of Economic Affairs（2006）Handbook of Detention Facilities Planning, http://www.wrap.gov.tw/
- Wikipedia, Curve fitting, http://en.wikipedia.org/wiki/Curve_fitting
- Wikipedia, Rectangle method, http://en.wikipedia.org/wiki/Rectangle_method
- Wikipedia, Simpson's rule, http://en.wikipedia.org/wiki/Simpson's_rule
- Wikipedia, Trapezoidal rule, http://en.wikipedia.org/wiki/Trapezoidal_rule
