1. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 A High Order Relativistic Particle Push Method for PIC Simulations M.Quandt, C.-D. Munz, R.Schneider

2. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Overview  Motivation  Mathematical Model  Results of Convergence Studies  Non Relativistic Motion in Time varying E-Field  Relativistic B-Field Motion  Relativistic ExB Drift  Conclusions and future Works

3. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Motivation - Coupled PIC/DSMC/FP PicLas-Code concept to study electric propulsion systems - High Order PIC method for the self-consistent solution of the Maxwell-Vlasov equations - For Consistency: High Order Lorentz solver which increase accuracy and efficiency

4. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Lorentz Equation of Motion - Truncated Taylor series expansion : - Mathematical model :

5. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Lorentz Equation of Motion - Recursive second order scheme for the velocity :

6. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Lorentz Equation of Motion - Recursive second order scheme for the velocity :

7. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Lorentz Equation of Motion - Recursive second order scheme for the velocity :

8. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Lorentz Equation of Motion - Recursive second order scheme for the velocity :

9. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Convergence Studies Set up of numerical experiments : - fixed final time - at compute the norm - experimental order of convergence - number of discretization points

10. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Lorentz factor - Variation in amplitude, phase shift and angular rate - All derivatives can be computed immediately - Parameters : Benchmark 1: Non-Relativistic Particle Motion

11. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Benchmark 1: Non-Relativistic Particle Motion The result of EOC are in good agreement with expected formel order. The 3D Lissajous trajectories; Line: exact; dots: numerical

12. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Lorentz factor gamma > 1 and constant in time - Particle trajectory in xy-plane with constant B-Field - Parameters of a positive charge and mass of electron Benchmark 2: Relativistic Particle Motion in B-Field

13. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Benchmark 2: Relativistic Particle Motion in B-Field - Initial Values of positive charge and mass of electron, positive B-field and velocity - Particle trajectory calculated with a third and fifth order scheme for 10 cycles - Particle trajectory deviates due to the error accumulation in time Particle trajectory with 160 intervals starts at (0,0). Line: exact; filled circles:third order scheme

14. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 Benchmark 2: Relativistic Particle Motion in B-Field Slope of the graphs correspond to the experimental order of convergence. Improved Solution: fifth order - Initial Values of positive charge and mass of electron, positive B-field and velocity - Particle trajectory calculated with a third and fifth order scheme for 10 cycles - Particle trajectory deviates due to the error accumulation in time Particle trajectory with 160 intervals starts at (0,0). Line: exact; filled circles:third order scheme

15. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Problem reduction through Lorentz transformation for - Same equation as previously but now in primed reference system - Back transformation yields to ExB motion Benchmark 3: Relativistic Particle Motion in ExB-Field Lorentz transformation with velocity into primed reference system.

16. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields - Negative drift velocity and clockwise rotation Benchmark 3: Relativistic Particle Motion in ExB-Field Third order approximation: visible deviation from the analytic solution.

17. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields - Negative drift velocity and clockwise rotation Benchmark 3: Relativistic Particle Motion in ExB-Field Expected orders of convergence reached with sufficent number points. Third order approximation: visible deviation from the analytic solution. Fifth order scheme: Perfect agreement between exact and numerical solution.

18. Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20, 2007 - Taylor series expansion in time applied to the Lorentz equation up to order 5 - Tested on 3 benchmark problems and EOC reached finally the expected order - Extension to higher order schemes (greater equal 6 ) - Stability analysis - Further benchmark tests -vs usual second order PIC scheme -vs defined problems of the HOUPIC project for accelerators - Coupling Maxell-Vlasov solver with FP and DSMC modules - Application to technical devices (pulsed plasma thruster, high-power high-frequency microwave generation) Conclusion and Future Works