Timeline for Best method to solve this system of PDEs?
Current License: CC BY-SA 4.0
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| Dec 6, 2021 at 17:08 | comment | added | Basham Govindan | Hey @PianoEntropy, oh yes when we discretize using n points we typically be inversing a (n-2)*(n-2) sparse square matrix. You can either vectorize or code using a simple for loops for writing down the equations and forming the matrix but I have inversed large matrices of the order of 5000*5000,which i believe is relatively small compared to kind of matrices which a low level langauge handles in a commerical system. I have observed if we don't explicitly calculate the inverse but instead solve the equations it's way faster ~ 5-10 secs for the sizes mentioned (in matlab) | |
| Dec 6, 2021 at 15:30 | comment | added | PianoEntropy | The problem with not having a time derivative for $v$ is that one can only use implicit methods to solve for $v$, but these have the problem that they scale poorly with mesh size (quickly becomes too cumbersome). | |
| Dec 6, 2021 at 14:46 | comment | added | PianoEntropy | So I've tried the finite difference method for an easier but related system, and it seems like the issue is that the equations quickly become too complicated for larger mesh sizes. Basically for e.g. 10 mesh points in space the system already needs to solve a 10x10 matrix and the expression become extremely. I used Mathematica as an algebraic solver, but maybe you know some way of dealing with this? | |
| Dec 6, 2021 at 14:30 | comment | added | PianoEntropy | Thanks! The equations are indeed analogous to advection diffusion reaction equations, but coupled to equations for linear elasticity. The third equation is force balance equation arising from some assumptions and is time-independent (i.e. there is no $\partial_t v$ anywhere). I hope this does not affect the rest of your answer? Yes, finite differences seem like a possible way forward. | |
| Dec 4, 2021 at 18:52 | history | edited | Basham Govindan | CC BY-SA 4.0 |
added 239 characters in body
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| S Dec 4, 2021 at 18:45 | review | First answers | |||
| Dec 4, 2021 at 20:43 | |||||
| S Dec 4, 2021 at 18:45 | history | answered | Basham Govindan | CC BY-SA 4.0 |