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See comments for the original discussion. The lack of L-stability of the trapezoidal rule in the first step is the source of your problem. A simpler and famous toy problem that shows the point is the Curtiss-Hirschfelder equation $$y'=-2000(y-\cos(t)) \\y(0)=0$$ Integrating this with Backward Euler and the Trapezoidal rule gives the following result (I wrote ...


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