ytic vs Numeric?

The use of computers to solve nonlinear (coupled) pde’s at this point in time is an invaluable resource. A good example is the Navier-Stokes equations which are hopeless to solve by another means. While new methods to solve these should be explored there is a simple reason why more time goes into improving old algorithms. With an older algorithm, people know it works and are confident in the results it produces. With a new algorithm the scientific community is skeptical and it requires a tremendous amount of validation to prove that this new algorithm gives the same results as older ones. Companies don’t have time for validation, they want to use methods they know work, and if you can speed up those old methods then that is even better.