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A comma splice run-on sentence uses a comma to separate two independent clauses, as in "I played chess, the boy watched".
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Generally use a comma to separate introductory phrases from the main clause (in this case, we needed one after the title).
In this paper, for convenience, we use a comma to denote partial differentiation, for example, (u_{,i}=frac{partial u}{partial x_{i}}), (u_{,ij}=frac{partial^{2}u}{partial x_{i}, partial x_{j}}).
In this section, we will use a comma to denote partial differentiation and adopt the summation convection, i.e., if an index is repeated, summation from 1 to N is understood, for example, u_{,i}u_{,k}u_{,ik}=sum _{i,k=1}^{N}frac{partial u}{partial x_{i}}frac {partial u}{partial x_{k}} frac{partial^{2} u}{partial x_{i},partial x_{k}}.
Use a comma to separate an absolute phrase.
Use a comma to separate an adverb clause.
Use a comma to separate dates and addresses.
Use a comma to indicate a break or pause within a sentence.
Use a comma to denote a break or pause within a sentence.
Use a comma to separate two or more adjectives describing a noun.
Use a comma to separate an introductory phrase from the rest of the sentence.
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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