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A Domain Decomposition Method for Pseudo-Spectral Electromagnetic Simulations of Plasmas J ean-Luc Vay, Lawrence Berkeley Nat. Lab. Irving Haber & Brendan Godfrey, U. Maryland PPPS 2013, San Francisco, CA, U.S.A. Office of Science. SciDAC - III ComPASS.

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A Domain Decomposition Method for Pseudo-Spectral Electromagnetic Simulations of Plasmas Jean-Luc Vay, Lawrence Berkeley Nat. Lab. Irving Haber & Brendan Godfrey, U. Maryland PPPS 2013, San Francisco, CA, U.S.A. Office of Science SciDAC-III ComPASS Particle-In-Cell method is ubiquitous for kinetic modeling of space and laboratory plasmas Charged particles (Newton-Lorentz) EM fields (Maxwell) Particle-In-Cell (PIC) PIC is the method of choice for simulations of plasmas and beams first principles includes nonlinear, 3D, kinetic effects, scales well to >100k CPUs and burns tens of millions of hours on U.S. supercomputers. Usual FDTD field solver scales well but impose serious limits on time step, accuracy, stability, etc. Spectral solvers do not scale well but offer higher accuracy and stability, eventually no time step limit (i.e. no Courant condition on field push). Analytic spectral solver for Maxwell’s equations of space and laboratory plasmas In the early 70s, it was shown by Haber et al1 that an exact solution exists for Maxwell in Fourier space if source J assumed constant over time step: “Pseudo-Spectral Analytical Time-Domain” or PSATD Note: taking first terms of Taylor expansion of C and S leads to pseudo-spectral formulation: “Pseudo-Spectral Time-Domain” or PSTD2 1I. Haber, R. Lee, H. Klein & J. Boris, Proc. Sixth Conf. on Num. Sim. Plasma, Berkeley, CA, 46-48 (1973) 2Similar to UPIC solver, V. Decyck, UCLA Spectral advantage on expansion of unit pulse of space and laboratory plasmas Kronecker d pulse FDTD PSTD PSATD Numerical dispersion, anisotropy, Courant condition: Numerical dispersion, isotropy, Courant condition: Exact dispersion, isotropy, Courant condition: None 1 But spectral method involves global operations: hard to scale to many cores. New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Example: unit pulse expansion at time T Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Separate calculation in two domains Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications guard regions Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Advance to time T+DT Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications spurious signal limited to guard regions Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Copy unaffected data Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations To affected areas Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations zero out remaining areas Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Ready for next time step Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New concept* enables scaling of spectral solvers of space and laboratory plasmas-- using property of Maxwell’s equations Ready for next time step Finite speed of light ensures that changes propagate a finite distance during a time advance enabling the use of local Fourier Tr. Replacing global “costly” local “cheap” by communications Successfully tested on 7x7 domain Hard to scale Easy to scale *J.-L. Vay, I. Haber, B. Godfrey, J. Comput. Phys.243, 260-268 (2013) New scheme successfully applied to modeling of of space and laboratory plasmaslaser plasma accelerators with Warp1 Lab frame Short laser propagates into long plasma channel, electron beam accelerated in wake. Improved phase space accuracy FDTD-PIC PSATD-PIC Warp-3D 0.5 0.5 gvz/c gvz/c spurious heating Artificial trapping correct behavior 0. 0. Warp-2D Lorentz boosted frame (wake) -0.5 -0.5 z(mm) z(mm) 0.1 0.1 0.11 0.11 Modeling in a boosted frame reduces # time steps2. Plasma drifting near C may lead to Num. Cherenkov. Improved stability FDTD-PIC PSATD-PIC Warp-3D instability Warp-2D 1A. Friedman, et al., PPPS 2013, paper 4A-3, Tuesday 6/18/2013, 17:00 Grand Ballroom B “Birdsall Memorial Session” 2J.-L. Vay Phys. Rev. Lett.98, 130405 (2007); 3B. B. Godfrey J. Comput. Phys.15 (1974) New scheme successfully applied to modeling of of space and laboratory plasmaslaser plasma accelerators with Warp1 Lab frame Short laser propagates into long plasma channel, electron beam accelerated in wake. Improved phase space accuracy FDTD-PIC PSATD-PIC Warp-3D 0.5 0.5 gvz/c gvz/c spurious heating Artificial trapping correct behavior 0. 0. Warp-2D Lorentz boosted frame (wake) -0.5 -0.5 z(mm) z(mm) 0.1 0.1 0.11 0.11 Modeling in a boosted frame reduces # time steps2. Plasma drifting near C may lead to Num. Cherenkov. Improved stability FDTD-PIC PSATD-PIC Warp-3D instability Warp-2D 1A. Friedman, et al., PPPS 2013, paper 4A-3, Tuesday 6/18/2013, 17:00 Grand Ballroom B “Birdsall Memorial Session” 2J.-L. Vay Phys. Rev. Lett.98, 130405 (2007); 3B. B. Godfrey J. Comput. Phys.15 (1974) Theory extended to 3-D by Godfrey confirms improved stability* Numerical energy growth in 3 cm, γ=13 LPA segment FDTD-CK simulation results included for comparison *B. B. Godfrey, et al., PPPS 2013, paper 4A-6, Tuesday 6/18/2013, 17:00 Grand Ballroom B “Birdsall Memorial Session” Broader range of applications stability* Electromagnetic MHD/Vlasov Heat equation Diffusion equation Vlasov equation General relativity Schrödinger equation any initial value problem(?) 1 domain 9 domains T=0 T=0 T=25Dt T=25Dt Challenge stability* -- novel method makes a small approximation Errors appear lower than standard PIC e.g., smoothing, guards cellsare effective Mix globalwith new localexchanges reduces further impact of approximation Relative error Test unit pulse grid 126×126 T=200Dx/(cDt) // runs: 3 CPUs standard spectral Dw/w0 4 guard cells 8 guard cells Typical PIC 16 guard cells Smoothing (bilinear, # passes) Future directions: error accumulations and mitigations will be studied in detail. Summary stability* Analytic spectral solver (PSATD) offers higher accuracy, stability & larger Dt New method uses finite speed of light to allow use of local FFTs enables scaling of FDTD with accuracy and stability of spectral Prototype implemented in 2-D in Warp Successfully tested on small unit pulse and laser plasma acceleration runs May be applicable to other initial value problems Small approximation could be an issue & further testings are underway

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