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A few things jump out at me from your code as potential problems.
You seem to be using Newton's method separately for the $y_1$ and $y_2$ variables.
This is not the same as using Newton's method for a nonlinear system involving both $y_1$ and $y_2$.
For the full Newton method, you'll be solving a 2x2 linear system at every step.
You'll have to calculate not ...
2
The series that converges to $\ln(2)$ appears to be suitable for Cohen-Villegas-Zagier acceleration [PDF]. This is an acceleration technique for alternating series, but continued fractions with positive positive partial numerators and denominators are equivalent to alternating series.
In particular, if $S_m$ is the $m$-th continued fraction approximant, then ...
2
You can de-singularize Riccati equations $y'=y^2+a$ by setting $y=-\frac{u'}{u}$ to get $u''+au=0$. If $a$ is continuous on the integration interval, the solution will be well-behaved on that interval. Solving with $u(0)=1$ and $u'(0)=-y(0)=-2$ gives the plot
with a root of $u$ and thus a singularity of $y$ at about $t=0.35230$. In any numerical integration,...
1
Look at the diffusion dispersion relations:
Upwind, ENO, WENO, see Fig. 2
DG, CG, see Fig. 15
Regards
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