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There are many kinds of RK methods which have extensions to exploit linearity. They all use some form of exponential or Lie Group idea (again exponential) to do so. Thus they generally do some form of integrating factor method and then apply the Runge-Kutta method to the IF transformed equation. If you want to see a listing of some of these, you can check ...


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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 ...


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Specific explanation: Consider the ODE $\dot{y}=f(t,y)$. The exact solution $y(t)$ verifies $$y_{n+1}=y_n + \int_{t_n}^{t_{n+1}}f(t, y(t)) dt$$ for each time step. To numerically compute the integral, one of the quadrature rules can be used, which consist of computing the weighted average of some selected function values on the domain of integration (see ...


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