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I have a book that quotes:

Euler's method, Modified Euler's method and Runge's method are Runge-Kutta methods of first, second and third order respectively. The fourth-order Runge-Kutta method is method is most commonly used and is often referred to as 'Runge-Kutta method' or 'classical Runge-Kutta method'

Similary Wikipedia categorizes Backward-Euler's method as ' Implicit methods' under the list of Runge-Kutta methods and also mentions:

The backward Euler method is first order.

Now the problem is that the same book (from which I have taken the above quote) solves the below problem using a method that seems quite different(at least to me) from the Backward-Euler's method.

Consider the first order initial value problem $y'=y+2x-x^{2}$,$y(0)=1$,$(0\le x\le\infty)$ with exact solution $y(x)=x^2+e^x$. For $x=0.1$, what is solution obtained using a single iteration of the second-order Runge-Kutta method with step size $h=0.1$

The book then shows the solution using:

$$k_1=hf(x_0,y_0)$$ $$k_2=hf(x_0+h,y_0+k_1)$$ $$y_1=y_0+\frac{1}{2}(k_1+k_2)$$

Here $f$ denotes the differential equation i.e. $y'=f(x,y)=y+2x-x^{2}$. Using the above equations and initial value, it gets the result as $y_1=1.1145$.

I tried to calculate the vaule using Backward-Euler's method using:

$$y_{1}=y_{0}+hf(x_{1},y_{1})$$ and I get the result as $y_1=1.1322$, which is different from the solution given in the book.


So I have the following questions:

  1. Is Backward-Euler method considered the same as Runge-Kutta $2^{\text{nd}}$ order (RK2) method? If yes, is my book incorrect with the solution?
  2. Is the method used in the book the actual Runge-Kutta $2^{\text{nd}}$ order method which is completely different from Backward-Euler's method?
  3. In case my first question's answer is yes, how can a method be a Runge-Kutta $2^{\text{nd}}$ order (RK2) while also being a $1^{\text{st}}$ order in itself? (no need to answer if first question's answer is no)

I am really confused with the way the book used the name Backward Euler as RK2 but then used a different method to solve a question that wanted RK2. Please help me understand this.

Note: My book states Backward Euler as Modified Euler's method (In case it's not so obvious).

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  • $\begingroup$ How did you find the $y_1$ to put into $f$? Whether they are the same or not depends on this initial calculation. $\endgroup$ – Paul Sep 7 '18 at 17:44
  • $\begingroup$ @Paul are you talking about the Euler's method ? $\endgroup$ – paulplusx Sep 7 '18 at 17:47
  • $\begingroup$ yes, the implicit Euler one. $\endgroup$ – Paul Sep 7 '18 at 17:47
  • $\begingroup$ @Paul if you use $y_1=y_0+h(y_1+2x_1-x_{1}^2)$ then you should be able to find $y_1$ as $h,x_1,y_0$ are all provided. $\endgroup$ – paulplusx Sep 7 '18 at 17:51
  • $\begingroup$ Your are then using forward Euler's to evaluate $y_1$ to put into $f$, but that is not the only way of determining the $y_1$ to put into $f$. $\endgroup$ – Paul Sep 7 '18 at 17:53
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No, the first order implicit backwards Euler method is different from the second order explicit trapezoidal or Heun method.

In your question you name and describe both methods correctly.

The list in the first book quote only refers to explicit methods, at no point is there a reference to the implicit backward Euler method.


To recap:

  • Forwards Euler method: $$y_{k+1}=y_k+h\,f(x_k,y_k)$$
  • Backwards Euler method: $$y_{k+1}=y_k+h\,f(x_{k+1},y_{k+1}),$$ which in general requires the solution of a non-linear equation.
  • Implicit trapezoidal method: $$\frac{y_{k+1}-y_k}h=\frac{f(x_k,y_k)+f(x_{k+1},y_{k+1})}2,$$ which again requires the solution of an in general non-linear equation. It is noted for its time symmetry.
  • Explicit trapezoidal method, modified Euler method, Heun's method: $$\frac{y_{k+1}-y_k}h=\frac{f(x_k,y_k)+f(x_{k+1},\tilde y_{k+1})}2,~~\text{ where }~~\tilde y_{k+1}=y_k+h\,f(x_k,y_k)$$ is a sufficiently accurate approximation of the implicit method. In the form of stages based on a Butcher tableau it is \begin{align}k_1&=h\,f(x_k,y_k),\\k_2&=h\,f(x_k+h,y_k+k_1),\\y_{k+1}&=y_k+\frac12(k_1+k_2).\end{align}
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  • $\begingroup$ But the book quotes as you see in the first paragraph Modified Euler's method and like I have written in my note it states Backward Euler as Modified Euler's method, I meant the implicit one. So is it safe to assume that the first quote(from my book) is wrong and the implicit Backward-Euler is not RK2 ? The list is from Wikipedia not from my book. $\endgroup$ – paulplusx Sep 9 '18 at 18:20
  • $\begingroup$ Where do you see that? The second quote names the second order Runge-Kutta method a.k.a. Heun method and the third quote gives the details of exactly that method. $\endgroup$ – LutzL Sep 9 '18 at 18:38
  • $\begingroup$ I am talking about the very first quote Euler's method, Modified Euler's method and Runge's method are Runge-Kutta methods of first, second and third order respectively. $\endgroup$ – paulplusx Sep 9 '18 at 18:47
  • $\begingroup$ All one-step methods with a Butcher tableau are Runge-Kutta methods. There are two commonly used second order RK methods, the midpoint method and the trapezium method. Also known in their explicit variants as improved and modified Euler methods. I assume that Runge's method is the shortened RK4 method. There is nothing wrong in your book quotes. $\endgroup$ – LutzL Sep 9 '18 at 20:18
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    $\begingroup$ Ok, so it is the title of that (sub-)section that has you stumped. And I agree, it is a bit disingenuous. However it makes sense to discuss the first order Euler methods before introducing improved or modified variants. Note that the methods used are properly named and nowhere is it claimed that they are second order or the announced modified Euler method. You should find that on the next pages. $\endgroup$ – LutzL Sep 10 '18 at 12:53

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