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A High Order Relativistic Particle Push Method for PIC Simulations

A High Order Relativistic Particle Push Method for PIC Simulations. M.Quandt, C.-D. Munz, R.Schneider. Overview. Motivation Mathematical Model Results of Convergence Studies Non Relativistic Motion in Time varying E-Field Relativistic B-Field Motion Relativistic ExB Drift

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A High Order Relativistic Particle Push Method for PIC Simulations

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  1. A High Order Relativistic Particle Push Method for PIC Simulations M.Quandt, C.-D. Munz, R.Schneider

  2. 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. 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. Lorentz Equation of Motion - Mathematical model : - Truncated Taylor series expansion :

  5. Lorentz Equation of Motion - Recursive second order scheme for the velocity :

  6. Lorentz Equation of Motion - Recursive second order scheme for the velocity :

  7. Lorentz Equation of Motion - Recursive second order scheme for the velocity :

  8. Lorentz Equation of Motion - Recursive second order scheme for the velocity :

  9. Convergence Studies • Set up of numerical experiments : • - fixed final time • - at compute the norm • experimental order of convergence - number of discretization points

  10. Benchmark 1: Non-Relativistic Particle Motion - Lorentz factor - Variation in amplitude, phase shift and angular rate - All derivatives can be computed immediately - Parameters :

  11. Benchmark 1: Non-Relativistic Particle Motion The 3D Lissajous trajectories; Line: exact; dots: numerical The result of EOC are in good agreement with expected formel order.

  12. Benchmark 2: Relativistic Particle Motion in B-Field - 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

  13. 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. 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 Improved Solution: fifth order Slope of the graphs correspond to the experimental order of convergence.

  15. Benchmark 3: Relativistic Particle Motion in ExB-Field - Problem reduction through Lorentz transformation for - Same equation as previously but now in primed reference system - Back transformation yields to ExB motion Lorentz transformation with velocity into primed reference system.

  16. Benchmark 3: Relativistic Particle Motion in ExB-Field - Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields - Negative drift velocity and clockwise rotation Third order approximation: visible deviation from the analytic solution.

  17. Benchmark 3: Relativistic Particle Motion in ExB-Field - Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields - Negative drift velocity and clockwise rotation Fifth order scheme: Perfect agreement between exact and numerical solution. Third order approximation: visible deviation from the analytic solution. Expected orders of convergence reached with sufficent number points.

  18. Conclusion and Future Works - 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)

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