EulerMethods.html

Numerical Solution of IVPs

Define an Initial Value Problem

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The solution is to be computed for the range and define .

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For this example, the exact solution can be expressed in closed form. This will allow us to compare our numerical solution with the exact result.

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Maple's default numerical solution

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Compare with the true solution at .

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Forward Euler Method

The Forward Euler Method can be derived from the Taylor series expansion of with local truncation error .

This is a method of order 1 because the global truncation error after steps where is: .

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Here is the differentiation operator so .

The resulting formula for the numerical method is: where .

First solve the IVP for the range with step size .

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Compare the computed solution with the true solution at .

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Plot the numerical values and the curve of the true solution.

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Note that the numerical values are off the curve near .

Next, solve the IVP for the range with the same step size .

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Compare the computed solution with the true solution at .

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Plot the numerical values and the curve of the true solution.

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The numerical solution is not very accurate.

This time, solve the IVP for the range with a smaller step size .

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Compare the computed solution with the true solution at .

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Plot the numerical values and the curve of the true solution.

Plot_2d

Modified Euler Method

For the Modified Euler Method, we start with one more term in the Taylor series expansion of . This time the local truncation error is .

This is a method of order 2 because the global truncation error after steps where is: .

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With and

we have the following potential numerical formula: .

While we know that, as before, we will have to approximate in some manner.

We will use the divided difference approximation: .

This leads to the following formula for the numerical method:

or, collecting terms

From the IVP we know that so we have developed the following implicit formula for :

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For the Modified Euler Method, we transform this into an explicit formula for via a two-step method.

Apply the Forward Euler Method to compute a first approximation for :

and then use this value in formula to get

The Modified Euler Method is one member of a class of methods known as Runge-Kutta methods.

Specifically, this is a Runge-Kutta method of order 2.

The formulas are often written in the following form.

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Restore the definition of for our particular IVP.

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Solve the IVP for the range with step size .

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Compare the computed solution with the true solution at .

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The numerical value at is more accurate than before.

Also, note that the value at the point is accurate to 3 significant figures this time.

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Plot the numerical values and the curve of the true solution.

Plot_2d

The computed solution follows the curve of the true solution much better with this method.

Now solve the IVP for the range with a smaller step size .

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Compare the computed solution with the true solution at .

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Plot the numerical values and the curve of the true solution.

Plot_2d

Test Order of Accuracy: ForwardEuler

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Compute successive ratios

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The ratio is approximately $Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn($ .

Conclusion: Since with we conclude that ForwardEuler is of order 1.

Test Order of Accuracy: ModifiedEuler

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Compute successive ratios

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The ratio is approximately $Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mn($ .

Conclusion: Since with we conclude that ModifiedEuler is of order 2.

Effects of Roundoff Error

Note that the last couple of values of are contaminated by roundoff errors in computing the numerical solution, so they are meaningless.

In fact, recall the number of steps used for computing the numerical solution that determined :

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We know from our study of "Floating Point Arithmetic" that adding numbers can incur a relative error of approximately . (This is in the "best case" where we are adding numbers all of the same sign.)

We are computing with Maple's default number of Digits and therefore is as follows:

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Therefore, the accumulated roundoff error when computing the numerical method with steps can be expected to be approximately:

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I.e., . Note that it is precisely at this level of accuracy in the values of that halving fails to yield improved accuracy.

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