Purpose

WaveToy is the simple test thorn that comes with Cactus as standard. This is written so that it solves the wave equation

$$\partial_t^2 \phi = \partial_{x^i}^2 \phi^i$$ (part541)

directly using the leapfrog scheme. This form of the equations isn't suitable for use with the method of lines.

The purpose of this thorn is to rewrite the equations in first order form

$$\partial_t \Phi = \partial_{x^i} \Pi^i,$$ (part542)

$$\partial_t \Pi^j = \partial_{x^j} \Phi,$$ (part543)

$$\partial_t \phi = \Phi,$$ (part544)

$$\partial_{x^j} \phi = \Pi^j.$$ (part545)

The first three equations (which expand to five separate PDEs) will be evolved. The final equation is used to set the initial data and can be thought of as a constraint.

This will be implemented using simple second order differencing in space. Time evolution is performed by the method of lines thorn MoL.