Cactus Code Thorn WaveToyCarpetX
Author(s)    : Erik Schnetter 
Maintainer(s): Erik Schnetter 
Licence      : LGPL
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1. Purpose

A wave toy using the CarpetX driver



2. First-order equations:

Scalar wave equation:
  phi,tt = phi,xx
Direct second order discretization:
  phi_0 = 2 phi_1 - phi_2 + dt^2 phi_1,xx

Introduce new variable:
  psi_0 = (phi_0 - phi_1) / dt

Some calculations:
  psi_1 = (phi_1 - phi_2) / dt
  phi_0 = 2 phi_1 - phi_2 + dt^2 phi_1,xx
  phi_0 = phi_1 + phi_1 - phi_2 + dt^2 phi_1,xx

  phi_0 = phi_1 + dt psi_1 + dt^2 phi_1,xx
  psi_0 = (phi_0 - phi_1) / dt

New set of equations:
  psi_0 = psi_1 + dt phi_1,xx
  phi_0 = phi_1 + dt psi_0



3. Scalar particles

Lagrangian:
  S = int 1/2 \eta^ab d_a u d_b u + 1/2 m^2 u^2 + \rho u

equation of motion:
  0 = dS/du = - \eta^ab d_a d_b u + m^2 u + \rho

nonsense: [0 = dS/d\rho = u]
nonsense: [0 = dS/dm = m u^2]

Ansatz:
  \rho = q \delta(x)

  0 = dS/dx = -q (grad u)(x)
  0 = dS/dq = u(x)