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Thoughts on time stepping and MHD

Thoughts on Energy Tracking in AstroBEAR

AstroBEAR does not track the thermal, magnetic, and kinetic energy separately (because it is trying to conserve total energy via conservative fluxing). It does however separately track density, momentum, and magnetic fields - which can be used to derive magnetic and kinetic energy - using thermal energy as a reservoir for discretization errors.

When you have flows that are dominated by non-thermal forms of energy ( \beta << 1 or M >> 1 ), discretization errors in total energy (while dynamically unimportant) can still lead to significant relative errors in thermal energy - which can be problematic if there are significant temperature-dependent microphysical processes. In those cases it might be better to solve the thermal energy equation independently and not worry about conserving total energy.

Thoughts on time-stepping

Simulations typically run for a few dynamical times - and for flows that are kinetic energy dominated ( M >> 1 and M_A >> 1 ) the computational time is independent of the flow speed (only a function of resolution N ). However, for flows that are magnetic or thermally energy dominated - the time stepping is limited by the Alfven or sound speeds respectively - and the computational time goes as \frac{1}{\min \left [M, M_A \right]} (ignoring the extra factor of N^D due to changes in the number of zones with resolution)

This can be combined as

T_c \propto \frac{1}{\min \left [1, M, M_A \right ]}

This makes simulating the RT instability M << 1 relatively computational expensive. Alfven waves can also restrict the time stepping when

M_A = \beta^{1/2} M < 1.

So simulations with modest M but very small \beta will also be relatively computational expensive.

High Alfven speeds are also somewhat easier to generate - since the magnetic fields/energy and density are not as tightly coupled as the thermal energy and density - due to material being free to leave along field lines - and the lack of flux freezing when magnetic resistivity is used.

Explicit Magnetic Resistivity

Astrobear currently implements magnetic resistivity explicitly - without subcycling - so time steps are limited by the smaller of the cell diffusion time \frac{\Delta x^2}{\eta} and the cell crossing time \frac{\Delta x}{U}. The cell diffusion time will be smaller than the cell crossing time when

\frac{\Delta x^2}{\eta} << \frac{\Delta x}{U} \rightarrow R_M < N

So the computational time to simulate a crossing time will go as

T_c = \frac{\max \left[1, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]}

So - you pay a penalty with explicit time stepping when R_M < N that could be avoided with implicit time stepping.

Simulations involving advection around and diffusion through an obstacle

Simulations involving advection around and diffusion through an obstacle will need to run for the longer of the crossing time around - and the diffusion time within (which is longer than the crossing time by a factor of R_M) - so in the explicit case we have

T_c = \frac{ \max \left[1, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]} \times \max \left[1, R_M \right] = \frac{\max \left[R_M, N, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]}

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