 Theta method

This one-step family interpolates between the Forward Euler method at $a=0$, the trapezoidal rule at $a=1/2$, and the Backward Euler method at $a=1$. Its stability function is $R(z)=[1+(1-a)z]/[1-az]$. It is first order except at one half, where it is second order. Applying it to spatial diffusion gives the theta diffusion stability criterion.