I had heard of a site called Emuparadise somewhere, so that is where I looked to find the game. You can download the Sonic Adventure 2: Battle and A LOT of other games from that site. The games you can download are called ROMs and the programs you need to play them are called Emulators. Here is a link to the Emuparadise site, which has links to the ROMs and a lot of good information about emulators:
Once you go to Emuparadise, click on “ROMs, ISOs & Games” from the site navigation. From that page, choose the console type you want to find games for. I chose “Nintendo Gamecube ISOs”. After you choose your console, you are taken to a page where you can either pick from the top games or search for a specific game listed in alphabetical order. I clicked “S” to search for the Sonic The Hedgehog games and then I clicked on the link for the Sonic Adventure 2: Battle game. The game page gives you a description of the game and some screenshots. Scroll down on that page and click the “Direct Download Link”. This might be a little confusing because clicking that link doesn’t download the game, but it takes you to another page where you can actually find the link to the game file. The actual game file will be under the “Direct Download” section of that page. Be very patient while the ROM is downloading. Some games can actually take hours to download from that site.
The file you will download will be a compressed file like a .rar or .7z file. Some emulators will uncompress the game when you want to play it, but the emulator I am going to write about makes you uncompress the file first. I use The Unarchiver or UnRarX to uncompress files.
Once you download the file and uncompress it, you will have a .iso file. This is the file that the emulator needs to be able to play the game. All of this sounds complicated, but it really isn’t. You can delete the compressed file once you have uncompressed it and have your .iso file.
Once you have installed Dolphin, run the program. The first time I ran it, it did not automatically find my Sonic Adventure 2: Battle .iso file. I had to click “Open” and find the .iso file. Since then though, every time I open Dolphin, my game is listed in the home screen. Here is a tip – to make your game full screen, press Command-F (Ctrl-F for PC). Otherwise your game plays in a small window. Another tip – some of the game files are very big, so you might want to only download the ones you want most. The Sonic Adventure 2: Battle file is over 1GB.
http://sonic.wikia.com/wiki/Sonic_Adventure_2. Spoiler alert! That page tells the story of the game if you want to play it before you read about it!