When trying to solve PDEs numerically, we employ methods like finite element method or method of lines. In method of lines we usually discretise in space and then solve the subproblems using a time-integration method for solving ODEs.
My question is, why is the time variable in a PDE given special treatment?
For instance:
- Couldn't you use space-time elements in the finite element method?
- The initital condition may be treated as a boundary condition
- In the wave equation: $\frac{\partial^2 u}{\partial t^2}=c\frac{\partial^2 u}{\partial x^2}$, we have a lot of symmetry in space and time. Why not treat them both as space?
Possible reasons for this may be:
- Physically it makes sense to somehow use an ODE solver, as evolution time is always dependent on the past, not the future, as we might have with forward finite differences.
- PDEs with half-infinite domains are difficult to solve numerically.
I understand that an answer to this question might not be a simple one, it may even be just convention. I checked for duplicate questions, but I might have missed one. Feel free to direct me to the appropriate question if such a question exists.