Cactus Code Thorn WaveToyCarpetX Author(s) : Erik Schnetter schnetter@gmail.com Maintainer(s): Erik Schnetter schnetter@gmail.com Licence : LGPL

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)