# Numerical Integration and the Runge Kutta Method

a) For a fixed delta t(i+1)= delta t, the linear equation, x = Ax, x(t0)=x0 has the solution... Show that the Runge Kutta method (below) is 3rd order and not 4th order for the above linear equation.

bi) Find the exact solution of x = -2x, x0 = 3

bii) Find the numerical solution for two simulation steps using explicit Euler and delta t=0.2.

biii) Find the numerical solution for two simulation steps using modified Euler and delta t=0.2.

biv) Find the global error at t=0.4 for the above methods. What do you expect will happen to these errors if the step size is halved?

See the attached file.

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#### Solution Summary

This solution is provided in 718 words combined and attached in two separate .doc files. It uses step-by-step equations for solving the mathematical problems regarding the numerical integration and uses the Runge Kutta method.