The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step:

$$ \frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{n+1}, t^{n+1}) + (1-\theta)f(u^{n}, t^{n})) = 0 $$

Some values of $\theta$ are often use and their pros and cons are well documented, notably:

  • $\theta = 0$ is purely explicit, and as such has a low computational cost
  • $\theta = 0.5$ (AKA the Crank-Nicolson method) has a precision in $O(\Delta t^2)$, vs $O(\Delta t)$ for other values
  • $\theta = 1$ is unconditionally stable for oscillations

I have also sometimes encountered the value $\theta = \frac{2}{3}$, referred as the Galerkin method, for example in [1] (p.1814), without any mention of its advantages over other values for $\theta$. The fact that the Galerkin method also refers to the much more popular method of transforming a PDE problem through a weak formulation doesn't help getting information.

Also, on a specific problem I'm working on, I find that the best results are obtained with a value of $\theta = 0.9$.

Do you know of any resources or have more info about the use non-typical values for $\theta$ in a theta-scheme? What are the possible advantages and caveats of such values?

[1] Dalhuijsen, A. J. and Segal, A. (1986), Comparison of finite element techniques for solidification problems. Int. J. Numer. Meth. Engng., 23: 1807-1829. doi:10.1002/nme.1620231003

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    $\begingroup$ Parameter choices for nonlinear problems are typically a crapshoot and very problem-specific so whatever answer you are looking for would probably have to do with applications related to your specific problem $\endgroup$ – whpowell96 Apr 13 at 4:09

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