Near 20 days before we update our website, we redesign and redevelop it, all the links are new, googel crawl our new website and it's include some new link on it's index but all the old links are still in google index. how these old links are remove on google index?
I think they cannot be removed but they can be pushed down, promote other pages of your site to bring them on top of those links
Old links should never be left to die. If you move the site, or if you restructure it so that the URLs change, you should always put a 301 redirect on all of the old pages, directing visitors and search engines to the correct new URL for each page. That will ensure that (a) anyone who has bookmarked a page on your site before the change still gets the right page, (b) anyone who has followed a link from another site that was set up before the change still gets to the right page, and (c) search engines will know that you have moved the page and will update their indexes much quicker.
Leaving the old URL giving a 404 error means that people and search engines are left in the dark. If I go to a URL that used to work and find that it just gives me a 404 error - and especially if that page has a message that says something along the lines of "We've recently redesigned our site, but we're just too damn lazy and ignorant to sort out the mess we've made of it, which is why you're seeing this error page" (OK, they aren't usually that honest), the chances are that I'll close it down and never go back.