There are many schools (grades K-12) around the country that offer some sort of science research option for students to pursue. Often this option is a local science fair, where students do smaller projects that are then presented at poster sessions. Many times the top local projects can advance to some type of regional and, possibly, state science fair. However, the projects students tend to do for a fair are "lower-level" research; that is, a project where the answer is at least known by the teacher and is often a lab activity from a standard experiment booklet. If there is a smaller subset of students who are more interested and motivated in science and who would like to take on more of a challenge with their research project, to the point where there is some element of originality and a truly unknown result being investigated (particularly in high school), most teachers do not have the means, or at least do not believe they have the means, of allowing them to do so unless there is a possibility of working in a university or professional laboratory outside of school. This more advanced level of research could be defined as the type that yields papers for major national science competitions. While the widespread belief among many high school teachers is that high-end research requires professional facilities, it is not always necessary to work outside of school in order to have a strong physics research program, especially when the research is computer based. This article offers suggestions for and examples of computational research projects for high school students, as well as students in introductory college physics classes.