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I've been learning about butcher tables and am having some difficulty understanding how to translate them when it comes to implicit methods. Specifically, I'm looking at backwards Euler:

\begin{array} {c|c} 1&1 \\ \hline & 1 \end{array}

Now generally, I am aware of how implicit methods work. A to-be-determined value lies on both sides of the equal sign and must be solved for algebraically:

$y_{n+1} = y_n + hf(t_{n+1}, y_{n+1})$

But what I am having a hard time figuring out is the notation when it comes to finding $k_i$ values from a butcher table in the implicit case.

What I got is this:

$$k_1 = hf(t_n + h, y + hk_1)\\ y_{n+1} = y_n + k_1$$

Am I to assume that $y + hk_1 = y_{n+1}$? If that is the case, how am I to understand the value of $k_1$ algorithmic-ally? It is an unknown, and appears twice. I am trying to learn how to turn Butcher tableaus into code, and this one is confusing me. The higher order explicit methods make sense to me though!

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To fix notation, denote the Butcher tableau by $$ \begin{array}{c|c} c & A\\ \hline & b^T \end{array} $$ where $b$ and $c$ are vectors of length $s$ (the number of stages) and $A$ is a $s \times s$ matrix.

Consider the ODE $$ y' = f(t, y) $$ and suppose that $y_n$ is a given approximation to $y$ at time $t = t_n$.

A general Runge-Kutta method (explicit or implicit) can be written as \begin{align} k_i &= f\left( t_n + \Delta t c_i, y_n + \Delta t \sum_{j=1}^s a_{ij} k_j \right), \tag{1} \\ y_{n+1} &= y_n + \Delta t \sum_{i=1}^s b_i k_i. \tag{2} \end{align} Advancing from $y_n$ to $y_{n+1}$ first requires finding the values $k_i$ for $1 \leq i \leq s$ using equation (1), and then computing $y_{n+1}$ using (2).

Note that if the method is explicit, then the matrix $A$ is strictly lower triangular, and each $k_{i}$ can be computed using $k_1, \ldots, k_{i-1}$ without solving a system of equations. On the other hand, if the method is implicit, then finding each $k_i$ will require solving a system of equations.

For concreteness, you give the example of backward Euler, which has the Butcher tableau $$ \begin{array}{c|c} 1 & 1\\ \hline & 1 \end{array} $$ There is only one stage, so substituting the values $c_1 = 1$ and $a_{11} = 1$ into equation (1), we obtain the following equation for $k_1$: $$ k_1 = f(t_n + \Delta t, y_n + \Delta t k_1). $$ As you mention, because $k_1$ also appears on the right-hand side of this equation, it will require solving an algebraic equation.

Once you have found $k_1$, you can compute $y_{n+1}$ using (2), which, after substituting $b_1 = 1$ gives $$ y_{n+1} = y_n + \Delta t k_1. $$

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  • $\begingroup$ How do you solve for it algebraically in an algorithmic sort of way? Isn't it going to depend entirely on the ODE and the initial conditions? For example, I tried $y' = y$ and got $k_n = \frac{1}{y_{n-1} - h}$, where h is my stepsize. Is it the job of the programmer to figure out this relationship for each new ODE he is trying to solve, or is there a way to automate this process? $\endgroup$ – rocksNwaves Nov 1 '19 at 0:05
  • $\begingroup$ For $y'= y$ you will get $k_1 = \frac{y_n}{1-\Delta t}$. Algorithmically, you can try to use a general-purpose solver to compute the solution. For example, you could use Matlab's fsolve where the objective function is $k_1 - f(t_n + \Delta t, y_n + \Delta t k_1)$. For more complicated examples it will require more work on the part of the programmer. $\endgroup$ – Will P. Nov 1 '19 at 0:17
  • $\begingroup$ The whole question is because I am trying to write my own code rather than using a generic solver. $\endgroup$ – rocksNwaves Nov 1 '19 at 0:22
  • $\begingroup$ It will depend on your ODE. Is it a scalar equation or is it a system? Is it linear or nonlinear? You can try to implement methods like Newton's method or bisection to solve the nonlinear equations. $\endgroup$ – Will P. Nov 1 '19 at 0:42

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