Energy equation solver

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The numerical integration is done in function AlgoGas, in SourceEuler.c. All functions or routines mentioned in this section are part of SourceEuler.c, unless otherwise stated. {{Source terms}} First, FARGO solves $\partial_t e = \rm{Source terms}$ . This is done in substep3. We briefly describe how this new substep is incorporated in AlgoGas. We denote with a

superscript the gas quantities at a given time

: (a) calculation of the sound speed with function ComputeSoundSpeed: $c_s^n = \sqrt{\gamma(\gamma-1) e^n / \Sigma^n}$ . This is necesary to calculate the timestep $\Delta t$ . (b) calculation of the pressure with function ComputePressureField: $p^n = (\gamma-1)e^n$ . This is necesary to solve Navier-Stokes equation. (c) substep1: the radial and azimuthal velocities

and $v_{\varphi}$ are updated with the source terms of the Navier-Stokes equation. This yields $v_r^{n+a}$ and $v_{\varphi}^{n+a}$ . (d) substep2:

, $v_{\varphi}$ and

are updated with some artificial viscosity. This yields $v_r^{n+b}$ , $v_{\varphi}^{n+b}$ and $e^{n+b}$ . (e) substep3:

is updated with the source term of the energy equation. This is done by a predictor-corrector scheme [[Stone, J. M., & Norman, M. L. 1992, ApJS, 80, 753]]:

$e^{\rm pred} = e^{n+b}+\Delta t(-p^n \vec{\nabla}\cdot\vec{v^{n+b}} + Q_{+}^{n+b}),$

with $p^n = (\gamma-1)e^n$ . Then:

$e^{n+c} = e^{n+b} + \Delta t(-\frac{1}{2}(p^n+p^{\rm pred})\vec{\nabla}\cdot\vec{v^{n+b}} + Q_{+}^{n+b}),$

where $p^{\rm pred} = (\gamma-1)e^{\rm pred}$ . {{2 - Advective transport}} The advective transport step is described in Stone & Norman (1992). We plot in the following figure the result of the 1D Sod test.

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