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Error Analysis Improved Euler Method Runge-Kutta Methods Runge Kutta Methods Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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! h 2 + ··· + y ( n ) ( x ) h n + y ( n + 1 ) ( c ) ( n + 1 ) ! h n + 1 . n ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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! h 2 + ··· + y ( n ) ( x ) h n + y ( n + 1 ) ( c ) ( n + 1 ) ! h n + 1 . n ! 2. Euler’s method . For y ′ = F ( x , y ) , y ( x ) = y 0 we use that y ( x )+ y ′ ( x ) ∆ x y ( x + ∆ x ) ≈ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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! h 2 + ··· + y ( n ) ( x ) h n + y ( n + 1 ) ( c ) ( n + 1 ) ! h n + 1 . n ! 2. Euler’s method . For y ′ = F ( x , y ) , y ( x ) = y 0 we use that y ( x )+ y ′ ( x ) ∆ x y ( x + ∆ x ) ≈ = y 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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! h 2 + ··· + y ( n ) ( x ) h n + y ( n + 1 ) ( c ) ( n + 1 ) ! h n + 1 . n ! 2. Euler’s method . For y ′ = F ( x , y ) , y ( x ) = y 0 we use that y ( x )+ y ′ ( x ) ∆ x y ( x + ∆ x ) ≈ = y 0 + F ( x , y 0 ) ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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! h 2 + ··· + y ( n ) ( x ) h n + y ( n + 1 ) ( c ) ( n + 1 ) ! h n + 1 . n ! 2. Euler’s method . For y ′ = F ( x , y ) , y ( x ) = y 0 we use that y ( x )+ y ′ ( x ) ∆ x y ( x + ∆ x ) ≈ = y 0 + F ( x , y 0 ) ∆ x = : y Euler ( x + ∆ x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x + ∆ x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x + ∆ x ) = y ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x y ( x + ∆ x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = = y Euler ( x + ∆ x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = y Euler ( x + ∆ x )+ y ′′ ( c ) 2! ( ∆ x ) 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = y Euler ( x + ∆ x )+ y ′′ ( c ) 2! ( ∆ x ) 2 = 4. So the error in each step is proportional to ( ∆ x ) 2 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = y Euler ( 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 steps gives an overall error ∆ x proportional to ∆ x . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Errors in Euler’s Method 3. But we know that y ( x )+ y ′ ( x ) ∆ x + y ′′ ( c ) 2! ( ∆ x ) 2 y ( x + ∆ x ) = y Euler ( 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 steps gives an overall error ∆ x proportional to ∆ x . (Details are more subtle than it looks.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods How can we Shrink the Error? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods How can we Shrink the Error? 1. Shrinking ∆ x is costly. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods

Error Analysis Improved Euler Method Runge-Kutta Methods Improving Euler’s Method logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Runge Kutta Methods