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logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Runge Kutta Methods

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 1. Taylor’s Formula. If the function y is n+1 times

differentiable, then for any h there is a c between x and x+h so that y(x+h)=y(x)+y′(x)h+y′′(x) 2! h2+···+y(n)(x) n! hn+y(n+1)(c) (n+1)! hn+1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 1. Taylor’s Formula. If the function y is n+1 times

differentiable, then for any h there is a c between x and x+h so that y(x+h)=y(x)+y′(x)h+y′′(x) 2! h2+···+y(n)(x) n! hn+y(n+1)(c) (n+1)! hn+1.

• 2. Euler’s method. For y′ = F(x,y), y(x) = y0 we use that

y(x+∆x) ≈ y(x)+y′(x)∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 1. Taylor’s Formula. If the function y is n+1 times

differentiable, then for any h there is a c between x and x+h so that y(x+h)=y(x)+y′(x)h+y′′(x) 2! h2+···+y(n)(x) n! hn+y(n+1)(c) (n+1)! hn+1.

• 2. Euler’s method. For y′ = F(x,y), y(x) = y0 we use that

y(x+∆x) ≈ y(x)+y′(x)∆x = y0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 1. Taylor’s Formula. If the function y is n+1 times

differentiable, then for any h there is a c between x and x+h so that y(x+h)=y(x)+y′(x)h+y′′(x) 2! h2+···+y(n)(x) n! hn+y(n+1)(c) (n+1)! hn+1.

• 2. Euler’s method. For y′ = F(x,y), y(x) = y0 we use that

y(x+∆x) ≈ y(x)+y′(x)∆x = y0 +F(x,y0)∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 1. Taylor’s Formula. If the function y is n+1 times

differentiable, then for any h there is a c between x and x+h so that y(x+h)=y(x)+y′(x)h+y′′(x) 2! h2+···+y(n)(x) n! hn+y(n+1)(c) (n+1)! hn+1.

• 2. Euler’s method. For y′ = F(x,y), y(x) = y0 we use that

y(x+∆x) ≈ y(x)+y′(x)∆x = y0 +F(x,y0)∆x =: yEuler(x+∆x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2 = yEuler(x+∆x)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2 = yEuler(x+∆x)+ y′′(c) 2! (∆x)2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2 = yEuler(x+∆x)+ y′′(c) 2! (∆x)2

• 4. So the error in each step is proportional to (∆x)2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2 = yEuler(x+∆x)+ y′′(c) 2! (∆x)2

• 4. So the error in each step is proportional to (∆x)2.
• 5. Summing the errors for b−a

∆x steps gives an overall error proportional to ∆x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Errors in Euler’s Method

• 3. But we know that

y(x+∆x) = y(x)+y′(x)∆x+ y′′(c) 2! (∆x)2 = yEuler(x+∆x)+ y′′(c) 2! (∆x)2

• 4. So the error in each step is proportional to (∆x)2.
• 5. Summing the errors for b−a

∆x steps gives an overall error proportional to ∆x. (Details are more subtle than it looks.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

• 1. Shrinking ∆x is costly.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

• 1. Shrinking ∆x is costly.
• 2. So a formula with a smaller error would be nice.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

• 1. Shrinking ∆x is costly.
• 2. So a formula with a smaller error would be nice.
• 3. The global error’s proportionality to ∆x in Euler’s method

came from the fact that Euler’s method uses the first two terms of the Taylor expansion.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

• 1. Shrinking ∆x is costly.
• 2. So a formula with a smaller error would be nice.
• 3. The global error’s proportionality to ∆x in Euler’s method

came from the fact that Euler’s method uses the first two terms of the Taylor expansion.

• 4. If we can capture more than the first two terms of the

Taylor expansion, we could get a global error proportional to (∆x)n.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

How can we Shrink the Error?

• 1. Shrinking ∆x is costly.
• 2. So a formula with a smaller error would be nice.
• 3. The global error’s proportionality to ∆x in Euler’s method

came from the fact that Euler’s method uses the first two terms of the Taylor expansion.

• 4. If we can capture more than the first two terms of the

Taylor expansion, we could get a global error proportional to (∆x)n. This would be good, because ∆x is usually small, so a higher power of ∆x would be even smaller.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2 y′′(x) = y′(x+h)−y′(x) h − y′′′(˜ c) 2! h

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2 y′′(x) = y′(x+h)−y′(x) h − y′′′(˜ c) 2! h y(x+h) = y(x)+y′(x)h+ y′′(x) 2! h2 + y

′′′(c)

3! h3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2 y′′(x) = y′(x+h)−y′(x) h − y′′′(˜ c) 2! h y(x+h) = y(x)+y′(x)h+ y′′(x) 2! h2 + y

′′′(c)

3! h3 = y(x)+y′(x)h+ 1 2h2 y′(x+h)−y′(x) h − y′′′(˜ c) 2! h

• + y

′′′(c)

3! h3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2 y′′(x) = y′(x+h)−y′(x) h − y′′′(˜ c) 2! h y(x+h) = y(x)+y′(x)h+ y′′(x) 2! h2 + y

′′′(c)

3! h3 = y(x)+y′(x)h+ 1 2h2 y′(x+h)−y′(x) h − y′′′(˜ c) 2! h

• + y

′′′(c)

3! h3 = y(x)+y′(x)h+ 1 2y′(x+h)h− 1 2y′(x)h− 1 2 y′′′(˜ c) 2! h3 + y

′′′(c)

3! h3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improving Euler’s Method

We’ll get y′′ from the expansion for y′. y′(x+h) = y′(x)+y′′(x)h+ y′′′(˜ c) 2! h2 y′′(x) = y′(x+h)−y′(x) h − y′′′(˜ c) 2! h y(x+h) = y(x)+y′(x)h+ y′′(x) 2! h2 + y

′′′(c)

3! h3 = y(x)+y′(x)h+ 1 2h2 y′(x+h)−y′(x) h − y′′′(˜ c) 2! h

• + y

′′′(c)

3! h3 = y(x)+y′(x)h+ 1 2y′(x+h)h− 1 2y′(x)h− 1 2 y′′′(˜ c) 2! h3 + y

′′′(c)

3! h3 = y(x)+ 1 2y′(x)+ 1 2y′(x+h)

• h+
• y

′′′(c)

3! − y′′′(˜ c) 4

• h3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x. The value yn will be an approximation for the value of the solution y at xn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x. The value yn will be an approximation for the value of the solution y at xn. (Formulated to make the transition to Runge-Kutta methods easier.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x. The value yn will be an approximation for the value of the solution y at xn. (Formulated to make the transition to Runge-Kutta methods easier.) The one step error is proportional to (∆x)3 and the global error is proportional to (∆x)2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method

For a given step length ∆x, with initial values (x0,y0), we compute the two recursively defined sequences {xn}∞

n=0 and

{yn}∞

n=0 via

xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x. The value yn will be an approximation for the value of the solution y at xn. (Formulated to make the transition to Runge-Kutta methods easier.) The one step error is proportional to (∆x)3 and the global error is proportional to (∆x)2. (Again we omit the considerable details.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Improved Euler Method versus Euler’s Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Reminder for Spreadsheet Implementation

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Reminder for Spreadsheet Implementation

Step length: ∆x. Initial values: (x0,y0). xn+1 := xn +∆x yn+1 = yn + 1 2k1 + 1 2k2, where k1 = F(xn,yn)∆x, k2 = F(xn +∆x,yn +k1)∆x. The value yn will be an approximation for the value of the solution y at xn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Its increment is f(xi)∆x 6 +4f

• xi+ 1

2

∆x 6 +f(xi+1)∆x 6 .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Its increment is f(xi)∆x 6 +4f

• xi+ 1

2

∆x 6 +f(xi+1)∆x 6 .

• 3. So how do we translate this into a formula for differential

equations that has high accuracy?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Its increment is f(xi)∆x 6 +4f

• xi+ 1

2

∆x 6 +f(xi+1)∆x 6 .

• 3. So how do we translate this into a formula for differential

equations that has high accuracy?

• 4. Match as many terms of Taylor’s formula as possible. The

remainder term gives the order of the error.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Its increment is f(xi)∆x 6 +4f

• xi+ 1

2

∆x 6 +f(xi+1)∆x 6 .

• 3. So how do we translate this into a formula for differential

equations that has high accuracy?

• 4. Match as many terms of Taylor’s formula as possible. The

remainder term gives the order of the error.

• 5. The global error in Simpson’s rule is ∼ (∆x)4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Further Improvements

• 1. The increment in the improved Euler method

1 2F

• x,y(x)
• ∆x+ 1

2F

• x+∆x,yEuler(x+∆x)
• ∆x looks like

the increment in the trapezoidal rule in numerical integration: 1 2

• f(xi)+f(xi+1)
• ∆x
• 2. Simpson’s rule is more accurate than the trapezoidal rule.

Its increment is f(xi)∆x 6 +4f

• xi+ 1

2

∆x 6 +f(xi+1)∆x 6 .

• 3. So how do we translate this into a formula for differential

equations that has high accuracy?

• 4. Match as many terms of Taylor’s formula as possible. The

remainder term gives the order of the error.

• 5. The global error in Simpson’s rule is ∼ (∆x)4, so we must

match the first four terms of the Taylor expansion.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0),

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

k4 = F(xn +∆x,yn +k3)∆x,

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

k4 = F(xn +∆x,yn +k3)∆x, yn approximates the value of the solution y at xn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

k4 = F(xn +∆x,yn +k3)∆x, yn approximates the value of the solution y at xn. One step error ∼ (∆x)5, global error ∼ (∆x)4.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

The most common Runge-Kutta Method

Step length ∆x, initial values (x0,y0), xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

k4 = F(xn +∆x,yn +k3)∆x, yn approximates the value of the solution y at xn. One step error ∼ (∆x)5, global error ∼ (∆x)4. (Considerable details omitted.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Fourth Order Runge-Kutta Method

y′ = y−x2, y(0) = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

Reminder for Spreadsheet Implementation

Step length: ∆x. Initial values: (x0,y0). xn+1 := xn +∆x yn+1 = yn + 1 6 (k1 +2k2 +2k3 +k4), where k1 = F(xn,yn)∆x, k2 = F

• xn + 1

2∆x,yn + 1 2k1

• ∆x,

k3 = F

• xn + 1

2∆x,yn + 1 2k2

• ∆x,

k4 = F(xn +∆x,yn +k3)∆x, yn will be an approximation for the value of the solution y at xn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta. Have seen a fourth order Runge-Kutta procedure.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta. Have seen a fourth order Runge-Kutta procedure. Setting up a formula for increments to match a Taylor polynomial actually leads to a system of equations for the parameters in the setup of the approximation formulas.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta. Have seen a fourth order Runge-Kutta procedure. Setting up a formula for increments to match a Taylor polynomial actually leads to a system of equations for the parameters in the setup of the approximation formulas. So there is more than one second order Runge-Kutta method.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta. Have seen a fourth order Runge-Kutta procedure. Setting up a formula for increments to match a Taylor polynomial actually leads to a system of equations for the parameters in the setup of the approximation formulas. So there is more than one second order Runge-Kutta method. Same goes for higher orders.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

logo1 Error Analysis Improved Euler Method Runge-Kutta Methods

General Runge-Kutta Methods

A jth order Runge-Kutta procedure computes an approximate solution to the differential equation y′ = F(x,y) by computing a sequence of values yn+1 := yn +a1k1 +a2k2 +···+amkm so that yn+1 agrees with the jth order Taylor polynomial of y at the previous evaluation point. The Euler method is a first order Runge-Kutta procedure. Improved Euler method: second order Runge-Kutta. Have seen a fourth order Runge-Kutta procedure. Setting up a formula for increments to match a Taylor polynomial actually leads to a system of equations for the parameters in the setup of the approximation formulas. So there is more than one second order Runge-Kutta method. Same goes for higher orders. Some of this freedom can be used to improve performance for certain types of equations.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Runge Kutta Methods