# Wed 20 Sept 06Spin Tracking Meeting1 Positron Capture Simulations: Runge-Kutta vs Boris Leo Jenner, Daresbury Labs.

## Presentation on theme: "Wed 20 Sept 06Spin Tracking Meeting1 Positron Capture Simulations: Runge-Kutta vs Boris Leo Jenner, Daresbury Labs."— Presentation transcript:

Wed 20 Sept 06Spin Tracking Meeting1 Positron Capture Simulations: Runge-Kutta vs Boris Leo Jenner, Daresbury Labs

Wed 20 Sept 06Spin Tracking Meeting2 Outline … Positron capture optics What is astra? Spin tracking… T-BMT equation Runge-Kutta integration method Boris method

Wed 20 Sept 06Spin Tracking Meeting3 Capture Optics Design Two coil magnet gives focussing solenoid field z z V.S.Kashikhin

Wed 20 Sept 06Spin Tracking Meeting4 ASTRA Klaus Floettmann

Wed 20 Sept 06Spin Tracking Meeting5 T-BMT Equation Gyromagnetic anomaly Lorentz Factor Differential Equation…

Wed 20 Sept 06Spin Tracking Meeting6 Let an initial value problem be specified as follows.initial value problem Then, the RK4 method for this problem is given by the following equation: where Thus, the next value (y n+1 ) is determined by the present value (y n ) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes:slope k 1 is the slope at the beginning of the interval; k 2 is the slope at the midpoint of the interval, using slope k 1 to determine the value of y at the point t n + h/2 using Euler's method;Euler's method k 3 is again the slope at the midpoint, but now using the slope k 2 to determine the y-value; k 4 is the slope at the end of the interval, with its y-value determined using k 3. Runge-Kutta Integrator Scheme Andriy Ushakov

Wed 20 Sept 06Spin Tracking Meeting7 Boris Numerical Integration Scheme 1.Advance w by vector term b a half-step 2.Advance w by M a full step 3.Advance w by b a half-step Jeff Smith

Wed 20 Sept 06Spin Tracking Meeting8 Boris: 2 nd order, good at preserving conserving quantities, 1 calculation per step R-K: 4 th order, 4 calculations per step (speed!) R-K vs Boris Will try both!

Wed 20 Sept 06Spin Tracking Meeting9 Conclusions … ASTRA can be used to simulate the motion of particles/bunches in the capture optics section …but doesnt have spin tracking included… Spin tracking subroutine using Boris integrator underway Spin tracking subroutine using Runge-Kutta integrator underway – cross-check Merlin?

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