Dear Yago, RK and AB methods have become popular because they are all-purpose schemes, which are well-suited for most generic problems. There exist schemes which fit specific classes of problems, and in such a specific setting may be more effective, e.g., BDF schemes in the case of stiff equations, or symplectic schemes for Hamiltonian systems. Do your problem fall within these classes?

Dear Yago

You can try the nonstandard finite difference (NSFD) method which has already proven to be effective.

Dear Yago Isasi,

You can look at also the compact finite difference methods. For example, a sixth-order one is presented in https://doi.org/10.1002/cnm.1349

Best regards,

Tahir Cosgun