ppc

Predictor Predictor Corrector scheme

from Kunz et al., 2014

Initially, all quantities are known at time step n. 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)

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)

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)