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Runge 2 nd Order Method. Civil Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Runge-Kutta 2 nd Order Method http://numericalmethods.eng.usf.edu. Runge-Kutta 2 nd Order Method. For.
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Runge 2nd Order Method Civil Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu
Runge-Kutta 2nd Order Methodhttp://numericalmethods.eng.usf.edu
Runge-Kutta 2nd Order Method For Runge Kutta 2nd order method is given by where http://numericalmethods.eng.usf.edu
y yi+1, predicted yi x xi+1 xi Heun’s Method Heun’s method Here a2=1/2 is chosen resulting in where Figure 1 Runge-Kutta 2nd order method(Heun’s method) http://numericalmethods.eng.usf.edu
Midpoint Method Here is chosen, giving resulting in where http://numericalmethods.eng.usf.edu
Ralston’s Method Here is chosen, giving resulting in where http://numericalmethods.eng.usf.edu
How to write Ordinary Differential Equation How does one write a first order differential equation in the form of Example is rewritten as In this case http://numericalmethods.eng.usf.edu
Example A polluted lake with an initial concentration of a bacteria is 107 parts/m3, while the acceptable level is only 5×106 parts/m3. The concentration of the bacteria will reduce as fresh water enters the lake. The differential equation that governs the concentration C of the pollutant as a function of time (in weeks) is given by Find the concentration of the pollutant after 7 weeks. Take a step size of 3.5 weeks. http://numericalmethods.eng.usf.edu
Solution Step 1: C1 is the approximate concentration of bacteria at http://numericalmethods.eng.usf.edu
Solution Cont Step 2: C2 is the approximate concentration of bacteria at http://numericalmethods.eng.usf.edu
Solution Cont The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as The solution to this nonlinear equation at t=7 weeks is http://numericalmethods.eng.usf.edu
Comparison with exact results Figure 2. Heun’s method results for different step sizes http://numericalmethods.eng.usf.edu
Effect of step size Table 1. Effect of step size for Heun’s method (exact) http://numericalmethods.eng.usf.edu
Effects of step size on Heun’s Method Figure 3. Effect of step size in Heun’s method http://numericalmethods.eng.usf.edu
Comparison of Euler and Runge-Kutta 2nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods (exact) http://numericalmethods.eng.usf.edu
Comparison of Euler and Runge-Kutta 2nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods (exact) http://numericalmethods.eng.usf.edu
Comparison of Euler and Runge-Kutta 2nd Order Methods Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results. http://numericalmethods.eng.usf.edu
Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/runge_kutta_2nd_method.html
THE END http://numericalmethods.eng.usf.edu