Semi-implicit method for ODEs?

[Awatarn]

I have two coupled ordinary differential equations:
$\displaystyle \frac{dx}{dt} = f(y) x$
$\displaystyle \frac{dy}{dt} = s(x) y$
To solve these equations, we generally use explicit method, but these equations are stiff equations. Therefore semi-implicit method might be a better choice.

I'm wondering if the following discretization mathematically legitimates or not?

$\displaystyle x^{n+1} = x^n + f(y^n) x^{n+1} dt$
$\displaystyle y^{n+1} = y^n + s(x^n) y^{n+1} dt$

The reason I do it this way is nonlinearity of $f(y)$ and $s(x)$.
Do you have any suggestion or recommended method?

 

[gtoguy97]

Yes, the discretization you propose mathematically legitimates. However, it is not a good choice for solving your system of equations since it is very likely that it will lead to numerical instabilities. A better option would be to use an implicit method such as the Backward Euler or the BDF method. These methods require the solution of a nonlinear system at each time step, which can be done using iterative methods such as Newton's method or fixed point iteration.