Numerical Relativity as a Tool for Studying the Early Universe

Table 2

Initial states for one-dimensional MHD tests^a.

Test	Left state	Right state	Grid	𝑡final
Fast shock (𝜇=0.2^b)	𝑢^𝑖=(25.0,0.0,0.0)	𝑢^𝑖=(1.091,0.3923,0.00)	𝑛=40	2.5
	𝐵^𝑖/√4𝜋=(20.0,25.02,0.0)	𝐵^𝑖/√4𝜋=(20.0,49.0,0.0)
	𝑃=1.0, 𝜌0=1.0	𝑃=367.5, 𝜌0=25.48
Slow shock (𝜇=0.5^b)	𝑢^𝑖=(1.53,0.0,0.0)	𝑢^𝑖=(0.9571,−0.6822,0.00)	𝑛=200	2.0
	𝐵^𝑖/√4𝜋=(10.0,18.28,0.0)	𝐵^𝑖/√4𝜋=(10.0,14.49,0.0)
	𝑃=10.0, 𝜌0=1.0	𝑃=55.36, 𝜌0=3.323
Switch-off fast rarefaction	𝑢^𝑖=(−2.0,0.0,0.0)	𝑢^𝑖=(−0.212,−0.590,0.0)	𝑛=150	1.0
	𝐵^𝑖/√4𝜋=(2.0,0.0,0.0)	𝐵^𝑖/√4𝜋=(2.0,4.71,0.0)
	𝑃=1.0, 𝜌0=0.1	𝑃=10.0, 𝜌0=0.562
Switch-on slow rarefaction	𝑢^𝑖=(−0.765,−1.386,0.0)	𝑢^𝑖=(0.0,0.0,0.0)	𝑛=150	2.0
	𝐵^𝑖/√4𝜋=(1.0,1.022,0.0)	𝐵^𝑖/√4𝜋=(1.0,0.0,0.0)
	𝑃=0.1, 𝜌0=1.78×10^−3	𝑃=1.0, 𝜌0=0.01
Alfvén wavec (𝜇=0.626^b)	𝑢^𝑖=(0.0,0.0,0.0)	𝑢^𝑖=(3.70,5.76,0.00)	𝑛=200	2.0
	𝐵^𝑖/√4𝜋=(3.0,3.0,0.0)	𝐵^𝑖/√4𝜋=(3.0,−6.857,0.0)
	𝑃=1.0, 𝜌0=1.0	𝑃=1.0, 𝜌0=1.0
Shock tube 1	𝑢^𝑖=(0.0,0.0,0.0)	𝑢^𝑖=(0.0,0.0,0.0)	𝑛=200	1.0
	𝐵^𝑖/√4𝜋=(1.0,0.0,0.0)	𝐵^𝑖/√4𝜋=(1.0,0.0,0.0)
	𝑃=1000.0, 𝜌0=1.0	𝑃=1.0, 𝜌0=0.1
Shock tube 2	𝑢^𝑖=(0.0,0.0,0.0)	𝑢^𝑖=(0.0,0.0,0.0)	𝑛=200	1.0
	𝐵^𝑖/√4𝜋=(0.0,20.0,0.0)	𝐵^𝑖/√4𝜋=(0.0,0.0,0.0)
	𝑃=30.0, 𝜌0=1.0	𝑃=1.0, 𝜌0=0.1
Collision	𝑢^𝑖=(5.0,0.0,0.0)	𝑢^𝑖=(−5.0,0.0,0.0)	𝑛=200	1.22
	𝐵^𝑖/√4𝜋=(10.0,10.0,0.0)	𝐵^𝑖/√4𝜋=(10.0,−10.0,0.0)
	𝑃=1.0, 𝜌0=1.0	𝑃=1.0, 𝜌0=1.0

aIn all cases, the gas satisfies the Γ-law EOS with Γ=4/3. For the first 7 tests, the left state refers to 𝑥<0 and the right state 𝑥>0.
^b𝜇 is the speed at which the wave travels.
^cFor the nonlinear Alfvén wave, the left and right states are joined by a continuous function separated by 0.5 units.