A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time integrations Andrew Myers [email protected] Applied Numerical Algorithms Group, Computational Research Division with Phillip Colella, Brian Van Straalen, Bei Wang, Boris Lo, Victor Minden Advanced Modeling Seminar Series NASA Ames Research Center https://arxiv.org/abs/1602.00747 November10th, 2016 2 Talk Outline • Background and motivation • Description of our numerical method • Implementation using the Chombo framework • Numerical results • Summary and future research directions A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 3 Talk Outline • Background and motivation • Description of our numerical method • Implementation using the Chombo framework • Numerical results • Summary and future research directions A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 4 The Vlasov-Poisson Equations @f @f @f + v E =0 @t · @x − · @v 2φ = ⇢ E = φ r − r • The Vlasov-Poisson equations describe the evolution of a charged, collisionless fluid in phase space. • The electric field is obtained by solving Poisson’s equation. • Applicable as a simplified model to plasma physics (space plasmas, accelerator modeling, controlled fusion) and to cosmology. A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 5 Eulerian Methods • Nonlinear advection equation in a high-dimensional space, can be solved by Eulerian techniques. • Advantages: – Strong body of theory on finite volume methods – Do not suffer from “noise” – High order methods exist (Vogman + 2014) • Disadvantages: – High cost of grids in high-dimensional (4D, 5D, or 6D) spaces A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 6 Particle Methods • Discretize system with set of Lagrangian interpolating points, P f(x, v,t ) q δ x x i δ v v i ini ⇡ p − p − p p X2P • Reduces problem to system of ODEs for particle trajectories: dqp =0 v p x (t) dt p dx p = v E p dt p dv p = E dt − p • Can reconstruct distribution at later times from (x p(t), v p(t)) A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 7 Particle Methods • Variety of methods - mainly differ in how the force is computed • PIC methods, in which the force solve is performed on an intermediate grid, are particularly popular. • Advantages: – Mathematically simpler. Reduces a PDE to a set of coupled ODEs. – Naturally adaptive - the particles go where more resolution is needed. – Usually lower computational cost A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 8 Disadvantages of Particle Methods • Less Wang+2011 mathematical guidance • Particle noise prevents convergence • Usually limited to 2nd order accuracy A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 9 The goal of this talk • Describe a particle method (PIC) that attempts to address these downsides. • Show how to all the PIC stages at 4th-order – Heart of this is really the interpolation kernels. • Describe a high-order remapping procedure to ameliorate particle noise while maintaining 4th order accuracy A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 10 Talk Outline • Background and motivation • Description of our numerical method • Implementation using the Chombo framework • Numerical results • Summary and future research directions A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 11 Overview of a Particle-in-Cell time step Start • Deposition: Particle charges are deposited onto mesh: n+1 q x i x ⇢n+1 = p W − p i ∆x ∆x p ! Deposition X ⇣ ⌘ 2nd order: Piecewise linear, Cloud-in-Cell interpolation • Field solve: force is computed on the mesh by e.g. Field Solve solving Poisson’s Equation w/ 2nd order finite differences. Interpolation • Interpolation: Force is interpolated back to particle positions using same kernel. • Particle Push: Particle positions and velocities are Particle Push updated. 2nd-order leapfrog. A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 12 High-order in space (interpolation kernels) ⇢ ⇢ 1 • Given solution at a set of points, 0 ... reconstruct in between • Functions W(x) need to be even, normalized, have compact support x x ⇢(x) W j − ⇢ ⇡ ∆x j j X ✓ ◆ x0 x1 ... A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 13 High-order in space (interpolation kernels) • Commonly taken to be one of the B-spline functions. Limited to 2nd Order. NGP CIC TSC 1 x , 0 x 1, 3 x 2, 0 x 1/2, 1, 0 x 1/2, 4 M2(x)= −| | | | 1 −3| | 2 | | M1(x)= | | 0 otherwise. M3(x)= x , 1/2 x 3/2, 0 otherwise. 8 2 2 ⇢ 0−| | otherwise.| | ⇢ < : A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 14 High-order in space (interpolation kernels) • Error analysis involves looking at the Fourier transform of W ( x ) , W˜ (k) • Can show that W ˜ ( k ) 1 must have a zero of order n at k =0 AND W ˜ ( k ) must − have zeros of order n at k =2 ⇡ , 4 ⇡ , ... to be O(n). sin (k/2) n • For the Basis spline of order n, W˜ (k)= k/2 ✓ ◆ • Fulfills the second requirement, but zero at k = 0 is only order 2. • Hence, all the basis splines interpolate with at best 2nd order accuracy • They do, however, have increasing degrees of smoothness which may be desirable when the particles are highly disordered. • See Schoenberg 1975, Monaghan 1985 for more details. A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 15 High-order in space (interpolation kernels) • However, it is possible to design a polynomial with the appropriate zeros in Fourier space and transform back into real space to obtain a W ( x ) with the desired properties. • For details, see Lo, Minden, and Colella 2016. A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time B a0 =1 W˜ q(k) q/2 1 − W (x)= a ( 1)pM (2p)(x), q 2p − q p=0 X B [ q/16 2,q/2] − High-order in space (interpolation kernels)Wq q =4, 6 3 x 2 x | 2| x |2| +1, x [0, 1], 3 − | | − | | 2 x 2 11 x W4(x)=8 | | + x | | +1, x [1, 2], − 6 | | − 6 | | 2 <> 0, x 5 x 4 5 x 3 5 x 2 x :> | | + | | + | | | | | | +1, x [0, 1], • 4-point stencil − 12 4 12 − 4 − 3 | | 2 x 5 3 x 4 25 x 3 5 x 2 13 x • Reproduces8 cubic |24| |8| + 24| | |8| 12| | +1, x [1, 2], W6(x)= − − − | | 2 > x 5 x 4 17 x 3 15 x 2 137 x functions exactly> | | + | | | | + | | | | +1, x [2, 3], < − 120 8 − 24 8 − 60 | | 2 • Continuous with 0, > discontinuous> first derivative :> k k =4 q A high-order accurate Particle-In-Cell method q for Vlasov-Poisson problems over long time [ q/2,q/2] − 17 High-order in space (field solve) • Replace 2nd-order finite difference approximation with 4th-order, centered differences: D 2 φi+2ed + 16φi+ed 30φi + 16φi ed φi+2ed φ = ⇢ − − − − = ⇢i − 12∆x2 r − d=1 X d φi+2ed +8φi+ed 8φi ed + φi+2ed E = φ E = − − − r i − 12∆x • Resulting system can be solved with a variety of methods (we used geometric multigrid). A high-order accurate Particle-In-Cell method for Vlasov-Poisson problems over long time 1 The Method The goal is to solve the following system of equations for the particle posi- tions and velocities x and v given the force F : x˙ = v v˙ = F (t, x). (1) 18 The standard, fourth-order Runge-Kutta method applied to this system High-ordergives, for thein specialtime case (RK4) of a force that does not depend on v: 1 xn+1 = xn + vn∆t + (k +2k ) ∆t2 6 1 2 1 The Method 1 vn+1 = vn + (kn +4k + k ) ∆t, (2) • Replace leapfrog with 4th-Order Runge-Kutta6 1 2method3 The goal is to solve the following system of equations for the particle posi- where tions and velocities x and v given the force F : k = F (tn,xn) 1 x˙ = v 1 1 1 k = F tvn˙ += ∆Ft,( xt,n x+). vn∆t + k ∆t2 (1) 2 2 2 8 1 ✓ ◆ The standard, fourth-order Runge-Kutta method1 applied to this system k = F tn + ∆t, xn + vn∆t + k ∆t2 . (3) gives, for the special3 case of a force that does not depend2 2 on v: ✓ ◆ Note the sequentialn+1 naturen ofn this algorithm;1 k must2 be computed before x = x + v ∆t + (k1 +2k21) ∆t k2, which must be computed before k36. An alternative is to extrapolate the forces from then previous+1 timen 1 step.n For example, the forces at the stages v = v + (k1 +4k2 + k3) ∆t, (2) corresponding to times tn ∆6 t, tn 1 ∆t, and tn were kn, kn, and kn. − − 2 1 2 3 whereDefining n n k1 = F (t ,x ) t (tn ∆t) • Requires 3 force solves per time step⌧ = (velocity− − independent, force), instead(4) of 1 n 1 n ∆1 t n 1 2 k2 = F t + ∆t, x + v ∆t + k1∆t a 2nd order interpolating polynomial2 that2 passes8 through the required points • Self-starting. Gives up on symplectic✓ property. ◆ is n n n 1 2 k3 = F t + ∆t, x + v ∆t + k2∆t . (3) n n n 2 n 2 n n n f˜(⌧)=(2k ✓4k +2k ) ⌧ +( 3k +4k ◆k ) ⌧ + k .