A Multi-Patch Wave Toy

Erik Schnetter <schnetter@aei.mpg.de>
Luis Lehner <lehner@lsu.edu>
Manuel Tiglio <tiglio@lsu.edu>

February 7, 2018

1 Physical System

The massless wave equation for a scalar field ϕ can be written as

μ(γμνd ν) = 0

where γμν = ggμν, and dμ μϕ. Using the expressions g = αh and

gμν = 1α2 βiα2 βjα2 γij βiβjα2 ,

where γij is the inverse of the three metric, the equation can be rewritten as

ϕ̇ = Π, (1) Π̇ = βi iΠ + α hi h α βiΠ + h α Hijd j + α hdit hβi α Πα ht h α , (2) di ̇ = iΠ (3)

with Hij α2γij βiβj. The non-shift speed modes with respect to a boundary with normal ni are

v± = λΠ + Hijn idj

and the shift speed modes are dA, with A transversal directions.

The physical energy is

E = 1 2 1 α Π2 + Hijd idj hdx3

and the way the equations above have been written this energy is not increasing in the stationary background case if homogeneous boundary conditions are given at outer boundaries, also at the discrete level (replacing above i by Di).

The implementation of the field equations in the code differs slightly from the above. Assuming a stationary background and expanding out the derivative operator, we obtain the equations in the final form

ϕ̇ = Π, (4) Π̇ = βi iΠ + α hi h α βiΠ + α hi h α Hij jϕ + α h h α Hij ijϕ. (5)

2 Non-linear addition to the multi-patch wave toy

Simple addition to the wave multi-patch toy to get started on this.

2.1 Physical System

This is just a non-linear wave equation obtained, a very minor modification to the wave equation thorn adding a different initial data and a slight modification to the right hand side. A reference to this is in a paper by Liebling to appear in Phys. Rev. D (2005). The non-linear wave equation is written as

u(γμνd ν) = ϕp

where γμν = ggμν, and dμ = μϕ and p must be an odd integer 3.

The initial data coded is given by

ϕ = Ae(r1R)2δ2 (6) Π = μϕ,r + Ω(yϕ,x xϕ,y) (7) di = ϕ,i (8)

with r̃2 = 𝜖xx2 + 𝜖yy2 + z2.

The parameters used for this initial data are given some distinct names to avoid conflicts with existing ones and are as follows:

Note, as the solution is not known, one must set for now the incoming fields to 0. CPBC might one day be put... though who knows :-)

3 Formulations

U = ρvxvyvz T = ρvi T (9) tU = Ai iU + (10) ||ni|| 1 (11)

3.1 dt

Setting ρ = tu.

RHS:

Hij = α2γij βiβj (12) tu = ρ (13) tρ = βi iρ + α 𝜖 i 𝜖 α βiρ + Hijv j (14) tvi = iρ (15)

Propagation matrix:

Ax = 2βx βxβx + α2γxx βxβy + γxyα2 βxβz + α2γxz 1 0 0 0 0 0 0 0 0 0 0 0 (16)

An = 2βin i βiβj + α2γij n i ni 0 (17)

Eigensystem:

λ1 = 0 , w1 = 0001 T (18) λ2 = 0 , w2 = 0010 T (19) λ3 = βx αγxx , w 3 = βx αγxxβxβx + α2γxxβxβy + α2γxyβxβz + α2γxz T (20) λ4 = βx + αγxx , w 4 = βx + αγxxβxβx + α2γxxβxβy + α2γxyβxβz + α2γxz T (21)

λt = 0 , wt = 0ti T (22) λ± = βin i ± αγij ni nj , w± = βin i ± αγij ni njHijnj T (23)

3.2 d0

Setting ρ = nu with na = Dat, leading to ρ = 0u = (1α)tu (1α)βiiu.

RHS:

Hij = γij (24) tu = αρ + βiv i (25) tρ = βi iρ + 1 𝜖i𝜖αHijv j + ρ 𝜖i𝜖βi (26) tvi = βj jvi + vjiβj + iαρ (27)

Propagation matrix:

Ax = βxαγxxαγxyαγxz α βx 0 0 0 0 βx 0 0 0 0 βx (28)

An = βini αγijni αni βknkδij (29)

Eigensystem:

λt = βin i , wt = 0 γijnjnktk + γjknjnkti T (30) λ± = βin i ± αγij ni nj , w± = ±γij ni njγijnj T (31)

3.3 dk

(This section very probably contains errors, say Erik on 2005-04-13.)

Setting ρ = ku with a “Killing” vector ka = txa, leading to ρ = (1α)tu.

RHS:

Hij = γij βiβj α2 (32) tu = αρ (33) tρ = βi α iαρ + 1 𝜖i𝜖 βiρ + αHijv j (34) tvi = iαρ (35)

Propagation matrix:

Ax = 2βxα βxβx α2 + γxxα βxβy α2 + γxyα βxβz α2 + γxz α 0 0 0 0 0 0 0 0 0 0 0 (36)

Eigensystem: