Word for something that "helps solve a problem"
Solution 1:
The word tool is often used metaphorically to refer to methods and software. For instance, a word processor is a common tool of writers and editors. And the Socratic Method is a tool employed by teachers.
See these definitions in ODO:
1.1 A thing used in an occupation or pursuit:
computers are an essential tool
the ability to write clearly is a tool of the trade1.3 Computing: A piece of software that carries out a particular function, typically creating or modifying another program.
Solution 2:
Although you have discounted it, I do think that device would work well here, perhaps with a qualifying, attributive adjective.
Likewise, I think that, whilst tool is used more commonly used in such circumstances, aid is also appropriate and would convey your meaning clearly.
The system requires a stiff solver and appropriate algorithmic device, such as the Jacobian Matrix Pattern.
A numerical device such as the Jacobian Matrix Pattern greatly assists in solving the problem.
An appropriate algorithmic device, such as the Jacobian Matrix Pattern, is an invaluable aid for computing the solution.