Constructing an implicit 2-step process of order 4

Question

Construct an implicit 2-step process of consistency order 4 and check its stability

As stated, I want to construct such a process.

I start by definition of an implicit 2-step process:

$\sum_{j=0}^2 a_jy_{k+j}=h\sum_{j=0}^2 b_jf(t_{k+j}, y_{k+j}), k=0,\dotso, N-2$

Where $b_2\neq 0$. I also wonder what $N$ is here supposed to be?

This gives me the following equation:

$a_0y_k+a_1y_{k+1}+a_2y_{k+2}=h(b_0f_k+b_1f_{k+1}+b_2f_{k+2})$

Now I need to find $a_0, a_1, a_2, b_0, b_1, b_2$.

For this I use the following criterion:

Let $a_k, b_k$ be coefficients of a multi-step process with characteristic polynomial $\rho$. Furthermore let $C_j=\sum_{k=0}^m a_k\frac{k^j}{j!}-\sum_{k=0}^m b_k\frac{k^{j-1}}{(j-1)!}$.

The process has consistency order $p$ iff $C_j=0$ for $j=1,\dotso, p$ and $\rho(1)=0$.

The characterisitc polynomial is given by:

$\rho(x)=a_0+a_1x+a_2x^2$

All in all I get the equations:

I) $a_0+a_1+a_2=0$

II) $a_1+2a_2-b_0-b_1-b_2=0$

III) $\frac12a_1+2a_2-b_1-2b_2=0$

IV) $\frac16a_1-\frac43a_2-\frac12b_1-2b_2=0$

V) $\frac1{24}a_1+\frac23a_2-\frac16b_1-\frac43b_2=0$

Now I have to solve this, but my problem is, that I have just 5 equations, but 6 variables I want to determine.

Have I done a mistake somewhere? Can you confirm my equations so far?

Also I want to use a solver for this system of equations. Can you recommand an online solver?

Thanks in advance.

Answer 1

As stated in the comments, with the help of LutzL I got the following solution:

First we solve the system of equations:

$\begin{pmatrix} 1&1&1&0&0&0&|0 \\ 0&1&2&-1&-1&-1&|0 \\ 0&\frac12&2&0&-1&-2&|0 \\ 0&\frac16&\frac43&0&-\frac12&-2&|0 \\ 0&\frac1{24}&\frac23&0&-\frac16&-\frac43&|0 \end{pmatrix}$

With the Gauß algorithm we get to this matrix:

$\begin{pmatrix} 1&1&1&0&0&0&|0 \\ 0&1&2&-1&-1&-1&|0 \\ 0&0&2&1&-1&-3&|0 \\ 0&0&0&-3&1&-1&|0 \\ 0&0&0&0&2&-8&|0 \end{pmatrix}$

For simplicity we set $b_2=1$ and solve for

$a_0=-3$, $a_1=0$, $a_2=3$, $b_0=1$, $b_1=4$, $b_2=1$.

So we construced the implicit 2-step process:

$-3y_k+3y_{k+2}=h(f_k+4f_{k+1}+f_{k+2})$

Now we check the stability. We use the following theorem:

Let $f$ be Lipschitz continuous in the second variable. Let a multi-step process with characteristical polynomial $\rho$ be given. Then holds: If $\rho$ fulfills the criterion of Dahlquist, then the process is stable.

The criterion of Dahlquist states that we have to check the solutions $\rho(x)=0$. For every solution $\lambda$ has to hold $|\lambda|\leq 1$ and if $|\lambda|=1$, then $\lambda$ is a simple root.

It is $\rho(x)=3x^2-3$ and $3x^2-3=0\Leftrightarrow (x-1)(x+1)=0$. So $x_{1,2}=\pm 1$.

Obviously the criterion of Dahlquist is fulfilled. So the process is stable.