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Computational Error Analyses for Euler's Method, Runge-Kutta

4th and 6th Methods

Brenton K. Jones

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Euler's Method

Uses the first derivative to obtain slope of tangent a tangent line

Then uses the slope to approximate the value of the solution

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Error for Euler’s method

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Runge-Kutta Method 1

k1 is the slope at the beginning of the interval;

k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point tn + h / 2 using Euler's method;

k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value;

k4 is the slope at the end of the interval, with its y-value determined using k3.

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Runge-Kutta Method 2Uses trial step at midpoint of an interval to cancel out lower order error termsEstimate is determined by weighted averages of slope at midpointsUsed to give numerical solutions to differential equations (approximations)Generally very accurate

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Differential Equation: y''+y=0Initial conditions: y(0)=1, y'(0)=0Exact Solution: y=cos t

The Solution of Harmonic Oscillator

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Errors from the theoretical solutionswith Euler’s, Runge-Kutta 4th, and Runge-Kutta 6th order.

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Error Comparisons

Euler’s vs Runge-Kutta 4th order Runge-Kutta 4th vs Runge-Kutta 6th order

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Step sizes .005, .05, .2

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Differential Equation: y''-2y'-3y=0Initial conditions: y(0)=1, y'(0)=2Exact Solution: 0.25 exp[-t]+0.75exp[3t]

The other example 1

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Errors from the theoretical solutionswith Euler’s, Runge-Kutta 4th, and Runge-Kutta 6th order.

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Interesting Results (For this exponential solution, the 4th and 6th have similar properties in terms

of the error.)

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Differential Equation: u''+16u'+192u=0Initial conditions: u(0)=0.5, u'(0)=0Exact Solution: ))28sin(2)28cos(2(

4

1)( 8 ttetu t

The other example 2

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Step size .005

Errors from the theoretical solutionswith Euler’s, Runge-Kutta 4th, and Runge-Kutta 6th order.

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Differential Equation: y*y''-(y')^2=1Initial conditions: y(0)=1, y'(0)=0Exact Solution: y=cosh(t)

The other example 3

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Step size .005

Errors from the theoretical solutionswith Euler’s, Runge-Kutta 4th, and Runge-Kutta 6th order.

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Conclusions

Euler's Method (the 1st order) is the least accurate.

Runge-Kutta methods are generally very accurate.

Higher orders provide greater accuracy. The step size greatly effects results. Unless the solution converges to a stable

solution, larger t causes larger error.