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"run up" is correct and usable in written English.
You can use it to refer to an activity that is done quickly or that has been done over a short period of time. For example, "He ran up the stairs in record time."
Dictionary
run up
verb
To hasten to a destination.
Exact(53)
Run up the middle?
Overdraft fees can run up to $40.
"We have run up substantial legal costs.
Any trying to run up the aisle.
They run up and down the floor.
Have you run up against that?
Similar(7)
Now the dollar run-up is wider.
He lost his run-up.
"He did the same run-up".
In the run-up to the announcement.
The price run-up has been thrilling.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com