# Use of non-typical values of $\theta$ in theta-methods

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  (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?

 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

• 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 – whpowell96 Apr 13 at 4:09