I have a 30-year 5% mortgage. Should I re-fi or just accelerate the payments?

I have a 30-year 5% mortgage, and I was thinking about re-financing to 15 years 3.875% simply to save money in the long run. Assume I can afford the higher monthly payment on a 15-year. But why would I want to refinance at all? Can I get the exact same savings in long-term interest if I increase my payments to match those of a 15-year loan? I guess this is mostly a math question.