Home > Legacy archive > Specific versions > FARGO-AD > Energy equation solver

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 n superscript the gas quantities at a given time t: (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 v_r 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_r, v_{\varphi} and e are updated with some artificial viscosity. This yields v_r^{n+b}, v_{\varphi}^{n+b} and e^{n+b}. (e) substep3: e 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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