An analysis of block Verlet timestepping and lagged force evaluations for the gravitational N−body problem by Bimali Jayasinghe, PhD A Dissertation In Mathematics and Statistics Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Dr. Katharine Long Chair of Committee Dr. Victoria Howle Dr. Giorgia Bornia Dr. Joshua Padgett Dr. Sanjaya Senadheera Dr. Mark Sheridan Dean of the Graduate School August, 2018 Texas Tech University, Bimali Jayasinghe, August 2018 ACKNOWLEDGEMENTS I would like to acknowledge and thank the following people who have supported me throughout my PhD program. First, and foremost I would like to thank my advisor, Dr.Katharine Long, for guiding me through the PhD project with her methodic work and enthusiasm. I also thank her for correcting the draft of thesis with great patience and for making useful comments on the language and style. I would like to express my gratitude towards committee members, Dr. Victoria Howle, Dr. Giorgia Bornia, Dr. Joshua Padgett, for their valuable comments and suggetions. I am extremely thankful to the Department of Mathematics and Statistics in Texas Tech University for giving me an opportunity to pursue my graduate studies. Last but not least, I would like to extend my sincere thanks to my family and my friends for their continuous love, support and caring. ii Texas Tech University, Bimali Jayasinghe, August 2018 TABLE OF CONTENTS ACKNOWLEDGEMENTS.......................... ii ABSTRACT.................................. iv LIST OF TABLES...............................v LIST OF FIGURES.............................. vi 1. INTRODUCTION..............................1 1.1 Gravitational N− body problem.................1 1.2 Numerical Methods in N Body Simulation...........4 2. NUMERICAL ORBIT INTEGRATION..................7 2.1 Hamiltonian and N− body problem...............7 2.2 Symplectic Integrators......................9 2.2.1 Modified(or Symplectic) Euler Integrator..........9 2.2.2 The Stormer-Verlet Integrator................ 11 2.2.3 Interpretation as splitting method.............. 12 2.3 Individual timesteps........................ 14 3. COMPUTATION OF POTENTIAL.................... 16 3.1 Motion Under Gravity...................... 16 3.2 Potentials of Spherical Systems.................. 17 3.3 Potential-density pairs for other systems............ 22 3.4 Potential from Functional Expansions.............. 23 3.5 Poisson solvers for N− body code................ 25 3.5.1 Direct Summation....................... 25 3.5.2 Tree Code........................... 26 3.5.3 Other Poisson Solvers..................... 27 4. DETAILED ANALYSIS OF BLOCK TIMESTEP Verlet WITH LAGGED EXPANSION COEFFICIENTS.................... 28 4.1 Careful description of Algorithms................ 29 4.2 Error Analysis with Exact Forces................ 33 4.3 Error Analysis with lagged Forces................ 43 5. NUMERICAL EXPERIMENTS...................... 48 5.1 Generation of Samples from DF................. 48 iii Texas Tech University, Bimali Jayasinghe, August 2018 5.1.1 Position Distribution..................... 49 5.1.2 Velocity Distribution..................... 51 5.2 Experiments............................ 53 6. CONCLUSION............................... 64 6.1 Future Work............................ 65 BIBLIOGRAPHY............................... 66 iv Texas Tech University, Bimali Jayasinghe, August 2018 ABSTRACT N-body simulations are widely used in astrophysics to model the behavior of stellar system. The Verlet integrator is one of the principle numerical methods used to solve the N−body problem. It is famous because of its significant features such as good numerical stability, time reversibility, preservation of the symplectic form on phase space and economy of memory. Since it is symplectic, it exactly solves an approximate Hamiltonian. The Verlet integrator preserves certain conserved quantities exactly, such as the total angular and linear momentum and the phase-space volume. Since the density in a stellar system varies with distance from its center, it is extremely inefficient to integrate all the stars with the same timestep, so an integrator should allow individual timestep for each star. The block timestep scheme will help to do this. We have analyzed both Verlet and block timestep Verlet method. We proved that block timestep Verlet method has GE, O(h2). The block timestep Verlet method has same advantages as in Verlet method except the symplecticity. We have seen that the block timestep Verlet method more efficient than regular Verlet method. We carry out experiments to check our analysis. When developing N− body simulation, the force calculation is one of the important factors. The best numerical integrator for a problem is evaluate the force cheaply. We introduced a method called lagged calculation using the functional expansion method. v Texas Tech University, Bimali Jayasinghe, August 2018 LIST OF TABLES 4.1 Algorithm of BTS Verlet method.................... 32 5.1 Algorithm of accept/reject method................... 52 5.2 Algorithm of Verlet integration..................... 54 5.3 Number of force evaluations....................... 63 vi Texas Tech University, Bimali Jayasinghe, August 2018 LIST OF FIGURES 4.1 The block timestep scheme, for a system with 4 classes of particles.. 31 5.1 Plummer sphere for 1000 stars 2D and 3D plot............. 51 5.2 Orbit of a single star........................... 55 5.3 Angular momentum of a single star................... 56 5.4 Total Energy of a single star....................... 57 5.5 The order of the Global Error I..................... 58 5.6 The order of the Global Error II..................... 59 5.7 The Total Energy of many stars..................... 60 5.8 The Total Energy and Angular Momentum of many stars....... 61 5.9 The GE of BTS Verlet method with several timestep......... 62 vii Texas Tech University, Bimali Jayasinghe, August 2018 CHAPTER 1 INTRODUCTION The N-body problem is the problem of predicting the motion of system of particles that are interacting with each other by a physical force such as gravity or electro- magnetism. The equation of motion can be classical or quantum mechanical. Some of these various N-body problems are solvable exactly for any value of N: for exam- ple, both the classical and quantum N-body problems with harmonic oscillator forces have exact solutions. When the \particles" are vortices interacting via fluid forces, the problem is exactly soluble when N ≤ 3. Usually, the N-body problem means the problem of N classical particles interacting via gravity; with N ≤ 2 this problem has well-known solutions. With N = 3, there are several exact solutions for periodic orbits discovered by Euler (1767), Lagrange (1772) and Poincar`e(1892-1899). An exact, convergent series solution to the 3-body problem for almost all initial conditions was given by Sundman in 1912. Sundman solved the problem for N = 3 with non zero angular momentum. Sundman's method was generalized by Wang[40] to the case of N = 3 or N > 3 with zero angular momentum. However, the Sundman and Wang series converge so slowly that they are useless for practical purposes. Therefore, for studies of the N-body problem in dynamics we resort to numerical calculations. The stellar system is a dynamical system of stars usually under the influence of gravity. It contains many stars: Open cluster (102 −104), Globular cluster (104 −106), Dwarf galaxy (109), Galaxy (108−1012). In stellar dynamics all masses are comparable and motions are highly irregular. The investigating the motion of the stars in stellar systems became an N-body problem and N−body simulations are commonly used in astrophysics. It is used to study processes of non-linear structure formation such as galaxy filaments and galaxy halos from the influence of dark matter and to study the dynamical evolution of star clusters. 1.1 Gravitational N− body problem Consider N point masses mi, i = 1; 2; ::::; N in an inertial reference frame in a three dimensional space R3 moving under the influence of mutual gravitational attraction. 1 Texas Tech University, Bimali Jayasinghe, August 2018 By Newton's law, the gravitational force felt on mass mi by a single mass mj, Fij, is Gmimj Fij = 3 (qj − qi) (1.1) kqi − qjk −11 3 −1 −2 where G = 6:67300 × 10 m kg s [39] is the gravitational constant, qi is the position vector of mass mi and kqj − qik is the Euclidean distance between the masses mi and mj. The equation of motion of the particles in a dynamical system can be obtained nd by summing the forces, Fij, over all the particles and applying Newton's 2 law of motion: 2 N d qi X Gmimj mi 2 = 3 (qj − qi) (1.2) dt kqi − qjk i=1;i6=j Mathematically, the N-body gravitational problem involves the solution of a system of 3N second-order differential equations (1:2). The system given by (1:2) difficult to integrate because of the following two factors [3]: 1. instability 2. the force on each mass depends on the position of all other stars therefore the time needed in calculating the force increases as the square of the number of particles being integrated These difficulties limit the usefulness of some integration methods, such as the Runge- Kutta method, for the numerical integration of the gravitational problem. The Runge- Kutta is very expensive for large systems. That leads to investigate new methods. Early exploration of the numerical methods was based on trial and error. A major advance was made by Aarseth[1] by introducing variable