REQUEST
A polluted reservoir exacerbated by large-scale earth disturbances and drainage overrun from a nearby mining company was found to have an unacceptable concentration of bacteria and other acidic pollutants. Government regulations prompted the company to initiate a complete study of the situation. We were called to study the particulars and asked to find acceptable levels of concentration of the pollutants as fresh water would replenish the reservoir. We were given starting conditions, such as, the differential equation that governs the concentration of the pollutant as a function of time (in weeks) and the initial concentration of pollutant in parts per volume.

Our task was to find concentration levels of the pollutant for specific time ranges (after 4 weeks, after 8 weeks, and so forth).

Note:
This exact problem was originally solved using Euler's Method and 2nd Order Runge-Kutta Methods. In most problems encountered in computational engineering, the fourth-order Runge-Kutta integration method represents an appropriate compromise between competing requirements of low truncation errors per step and a low computational speeds. Basically, this method provides more accurate results than Euler's method does and the second-order Runge-Kutta method is still not used often in numerical applications.

Why then use Euler's Method or 2nd Order Runge-Kutta methods? The choice generally comes to the degree of acuracy required. For our reservoir problem Euler's Method was appropriate. Processing through the other 2 methods reinforced the degree of accuracy of each one.

SOLUTION
To numerically solve the problem we applied the first-order differential equation using Fourth-Order Runge-Kutta Method (illustrated as sample model of execution under 4th ORDER RUNGE-KUTTA METHOD and IMPLEMENTATION) to solve the problem . We set the initial step size of 2.5 weeks and found approximate concentration of pollutant for different time periods. A report listing levels of pollutant concentrations was printed as wells as a graphical representation of the findings.

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           static void TestRungeKutta4();
              {
                 double h = 0.001;
                 double x0 = 0;
                 double y0 = 1.0;
                 double result = y0;
                 double trct = 0;
                 for (int i = 0; i < 11; i++)
                 {
                    double x = 0.1 * i;
                    result = ODE.RungeKutta4(f, x0, result, h, x);
                    exact = 2.0 * Math.Exp(x) - x - 1.0;
                    double error = ((exact - result) / exact) * 100000;
                    int r = (int)(error * 10000);
                    trct = r / 10000.0;
                    listBox1.Items.Add(" " + x );
                    listBox2.Items.Add(" " + result);
                    listBox3.Items.Add(" " + exact);
                    listBox4.Items.Add(" " + trct);
                    x0 = x;
                 }
                 this.Text = "Fourth-Order Runge-Kutta Method / KeystoneMiningPost";
              }