Abstract4
Introduction5
1.0Theory of the N-Body Problem 7
1.1The Background and History of the N-body Problem7
1.2Newtonian Physics8
1.2.1An Example of Newtonian Physics With Just Two Bodies9
1.2.2An Example of Newtonian Physics With Four Bodies 10
1.2.3A Detailed Example of Newtonian Physics With Two Bodies 10
1.3Development of Methods to Solve the N-body Problem17
1.4What Is the "Best" Method? 20
1.4.1The Efficiency of a Method20
1.4.2The Accuracy of a Method21
1.5Types of Deviations In Star Movement 23
1.5.1Gaining or Losing Energy23
1.5.2Distortion23
1.5.3Perihelion Shift23
1.5.4Overstep Phenomenon24
1.5.5Slingshot Effect 25
1.5.6Program Bugs 26
1.6Speed-up Techniques 27
1.7The Error Term 27
2.0Types of N-body ODE Integration Methods30
2.1One Step Methods30
2.1.1Euler's Method (-m euler1)30
2.1.2Taylor Series (not implemented) 31
2.1.3Runge-Kutta's Method (-m rk4)31
2.2Multistep Methods 33
2.2.1Modified Taylor Series Method (-m taylor3) 34
2.2.2Adam-Bashford's Method (-m ab7 and -m ab4) 34
2.3Predictor-Corrector Methods 35
2.3.1Modified Euler's Method (not implemented)36
2.4Other formulas (Mid-point method) (none are implemented)37
2.5The Gragg-Bulirsch-Stoer Method (-m gpemce8)38
3.0Variable Step Size N-Body ODE Integration Methods 40
3.1Method of Changing the Step Size 41
3.2Method of Changing Orders 41
3.3Method of Changing the ODE Integration Method41
3.4Method of Internal Checking42
4.0Efficient Force Function Evaluation Methods 43
4.1Floating Point Rounding Errors and the Force Function44
4.2Distant Stars Can Be Grouped Together45
4.3Closely Packed Stars Can Be Moved at the Same Time 46
4.4Replace the Force Functions With a Potential Mesh 47
4.5Creating Tree Structures For Evaluating the Force Function 48
4.6Hybrid Force Evaluation Methods49
5.0Analysis of the ODE Integration Methods 51
6.0Conclusion 55
References56