Numerical PDE have many applications in artificial intelligence, Engineering, Physics, Earth Sciences, Bioinformatics and Earth Science. In short, it is a fundamental ingredient of all modern sciences.

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You can look at the fluid dynamics studies and see how important it is. This is just small portion of the application of the PDE in the real life problems.

The list of actual applications of nonlinear PDEs of extreme relevance and practical importance is really long. One example that is in plain sight is dynamic meteorology and numerical weather forecasting: the weather report you see every night on TV has been obtained from the numerical solution of a complex set of nonlinear PDEs. More about this specific topic in this book

https://doi.org/10.1016/C2009-0-63394-8

Applications of nonlinear PDEs has a long list ranging from science to engineering.

I would say that mathematics plays an important role in practically any technological or scientific achievement. Maybe it sounds extreme, but I am not aware of any exception. A good deal of these achievements is involving ordinary and partial differential equations. The mathematical models range from scalar, ordinary differential equations to complex systems of coupled, nonlinear partial differential equations, depending on how complex and accurate they have to be. If you think about numerical simulations supporting the development of a product, these are mostly nothing but numerical approximations for the solution of a mathematical model.

I would answer the question differently: given a certain technological or scientific achievement, which nonlinear differential equations were supporting its development? :)