I am trying to understand the validity of why we can theta discretize the solution to a PDE. For a PDE following: $$0 = \partial_tf + A f$$ I understand that for one discrete time step the solution to above would be $$f(t_h) = e^{\Delta t A} f(t_{h+1}) = \sum_{k=0}^\infty \frac{(\Delta t A)^2}{k!} f(t_{h+1}),$$ using a Taylor expansion and $\Delta t =t_{h+1} - t_h$. Then my question is, how can we see that the theta scheme solves above: \begin{align} f(t_{h+1/2}) &=[I+(1-\theta)\Delta t A] f(t_{h+1})\\ [I-\theta\Delta t A]f(t_h) &= f(t_{h+1/2}). \end{align}



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.