PBKDF2WithHmacSHA512 Vs. PBKDF2WithHmacSHA1?
Hash algorithm strength is important, but it is not so important in key derivation functions. It is unlikely that even if SHA-1 is broken that it would influence the security of PBKDF2. You are better off using SHA-1, and increase the iteration count up to a level that is tweaked for your specific configuration.

If you must, you could use Bouncy Castle to create an implementation of PBKDF2 with SHA-256. The internal security of the algorithm is not that important, if your server is relatively safe against side channel attacks (and for PBKDF2 it likely is). So using a non-certified open source solution would probably not compromise your security.

Your efforts are probably better spend elsewhere. Congratulate yourself on choosing a relatively good password based KDF, choose a 64 bit random salt and use a higher iteration count, and move on...



Note: if you want to protect against hardware accelerated attacks, you may want to consider scrypt instead of PBKDF2.



The Oracle JDK 1.8 contains PBKDF2 implementations for SHA-512 "PBKDF2WithHmacSHA512". This can be used for compatibility with other systems or for deprecating SHA-1 entirely.