Abstract Energy Error Analysis of Particle-In-Cell Simulations With Different Numerical Methods Author：Shao-Wei Hong Advisor：S W Y Tam Institute of Space and Plasma Sciences National Cheng Kung University SUMMARY From our simulations the results show the Velocity Verlet method RK2 method and RK4 method all have a good accuracy for energy conservation However we also observed a larger numerical error during the instability process Although the error of total energy compared with the variation of energy change between the kinetic energy and the electrical energy is negligible the numerical deviation would affect the subsequent numerical results One needs to reduce the error as much as possible Our simulation results show that using higher order numerical methods to approximate the momentum equation and reducing the time step can decrease the error during the plasma instability process Key word：plasma instability energy conservation Particle-In-Cell (PIC) method INTRODUCTION The Particle-In-Cell (PIC) method can be used to simulate the instability process in plasma physics The limitations of the numerical method would cause the numerical errors to increase Since the plasma instability process is associated with intense energy exchange and must satisfy the energy conservation we record the total energy of the system at every time step to evaluate the size of the numerical errors MATERIALS AND METHODS We examine different numerical methods for their accuracy in PIC simulations: Euler method Velocity Verlet method RK2 method and RK4 method Spatially we compare second-order fourth-order and sixth-order central finite-difference method for the Poisson equation RESULTS AND DISCUSSION We use the Particle-In-Cell method to simulate the bump-on-tail instability process and the results show an intense energy change during the instability process: the particles lose energy and wave energy grows and we can see the bump in the velocity distribution being smooth out These results correctly correspond to plasma physics Moreover we use the Fourier transform to get the frequency band and the wave number of the growing waves and use the relation v=ω/k to calculate the corresponding phase velocity in the velocity distribution We found that the range of phase velocities corresponds to the velocity range of positive slope so the PIC method can correctly simulate plasma instability From simulations we know there is a larger numerical error during the instability process and the error deviation can be reduced as we improve the accuracy of the approximation for any equation or reduce the time step CONCLUSION In our study the Velocity Verlet method RK2 method and RK4 method are numerically stable and we have to reduce all errors as much as possible to avoid the accumulation of errors affecting the subsequent numerical accuracy significantly Improving the order of numerical method and reducing the time step are both effective ways to reduce the numerical errors However practically to improve the order of numerical method is a more effective way to improve the accuracy of the entire system because to reduce the time step increase the overall number of steps of the calculation causing the round-off error to occur more often
Energy Error Analysis of Particle-In-Cell Simulations With Different Numerical Methods
紹惟, 洪. (Author). 2019
Student thesis: Doctoral Thesis