ppc

Predictor Predictor Corrector scheme

from Kunz et al., 2014

Initially, all quantities are known at time step n, that is E, B, J, N, V, x, v. Uppercase are for fields, lowercase are for particles quantities. In 3 main steps :

Predictor-1 :

• Bp1[n+1] with Maxwell-Faraday using B[n] & E[n]
• Jp1[n+1] with Maxwell-Ampère using Bp1[n+1]
• Ep1[n+1] with Ohm's law using Bp1[n+1], N(n), V(n), Jp1(n+1)

and then build E & B at n+1/2

• Bp1[n+1/2] = 1/2 (B[n]+Bp1[n+1])
• Ep1[n+1/2] = 1/2 (E[n]+Ep1[n+1])

so the particles can be pushed at n+1

• vp[n+1] with Newton's law using Bp1[n+1/2], Ep1[n+1/2] and v[n]
• xp[n+1] with motion eq. using vp[n+1] and x[n]

then moments are deposited on the grid

• Vp[n+1] with shape fonction and vp[n+1]
• Np[n+1] with shape fonction and xp[n+1]

Predictor-2 :

• Bp2[n+1] with Maxwell-Faraday using B[n] & Ep1[n+1/2]
• Jp2[n+1] with Maxwell-Ampère using Bp2[n+1]
• Ep2[n+1] with Ohm's law using Bp2[n+1], Np[n+1], Vp[n+1], Jp2[n+1]

and then build E & B at n+1/2

• Bp2[n+1/2] = 1/2 (B[n]+Bp2[n+1])
• Ep2[n+1/2] = 1/2 (E[n]+Ep2[n+1])

Corrector :

• B[n+1] with Maxwell-Faraday using B[n] & Ep2[n+1/2]
• J[n+1] with Maxwell-Ampère using B[n+1]

so the particles can be pushed at n+1

• v[n+1] with Newton's law using Bp2[n+1/2], Ep2[n+1/2] and v[n]
• x[n+1] with motion eq. using v[n+1] and x[n]

then moments are deposited on the grid

• V[n+1] with shape fonction and v[n+1]
• N[n+1] with shape fonction and x[n+1]

and finally get the electric field

• E[n+1] with Ohm's law using B[n+1], N[n+1], V[n+1], J[n+1]