### Corrected some typos in the theory PDF.


Former-commit-id: 7fbf7981fa52d1c13d62cdd564c46d88f167b594
parent d5dfe1b3
 ... ... @@ -18,7 +18,8 @@ Every particle contains the following information: \begin{table}[h] \centering \begin{tabular}{|l|l|c|l|} Quantity & Type & Symbol & Unit \\ \hline \textbf{Quantity} & \textbf{Type} & \textbf{Symbol} & \textbf{Units} \\ \hline \hline Position & Primary & $\vec{x}$ & $[m]$ \\ Velocity & Primary &$\vec{v}$ & $[m\cdot s^{-1}]$ \\ ... ... @@ -41,19 +42,21 @@ Every particle contains the following information: \end{tabular} \end{table} Secondary quantities are computed from the primary one in a loop of particle neighbors. Tertiary ones are computed from seondary ones in another loop. \\ Secondary quantities are computed from the primary one in a loop (density loop) over all particle neighbors. Tertiary ones are computed from secondary ones in another loop (force loop). \\ For optimization purposes, any function of these quantities could be stored. For instance, $1/h$ instead of $h$ or $\frac{P}{\rho\Omega}$ instead of $\Omega$ may be options worth exploring. \\ The four quantities in the second half of the table are used in improved state-of-the-art implementations of SPH. In a first approximations, they can be neglected. The four quantities in the second part of the table are used in improved state-of-the-art implementations of SPH. In a first approximation, they can be neglected. \\ In what follows, we will use $\vec{r}_{ij} = \vec{x_i} - \vec{x_j}$ and $\hat{r}_{ij} = \vec{r}_{ij}/x|\vec{r}_{ij}|$ to simplify the notation. \section{Kernel function} In what follows, we will use $\vec{r}_{ij} = \vec{x_i} - \vec{x_j}$ and $\hat{r}_{ij} = |\vec{r}_{ij}|$. The kernel function can always be decomposed as: The kernel function can always be decomposed as: \begin{equation} W(\vec{x}, h) = \frac{1}{h^3}f\left(\frac{|\vec{x}|}{h}\right) ... ... @@ -128,14 +131,15 @@ where $\eta \approx 1.2$ is a constant. These two equations can be solved iterat bisection scheme. In practice, the loop is performed over all particles $j$ which are at a distance $|\vec{r}_{ij}|<\zeta h$ from the particle of interest. One has to iterate those two equations until their outcomes are stable.\\ Another measure of the accuracy of $h$ is to use the weighted number of neighbors which (in 3D) reads Another measure of the accuracy of $h$ is the weighted number of neighbors which (in 3D) reads \begin{equation} N_{ngb} = \frac{4}{3}\pi \left(\zeta h\right)^3 \sum_j W(\vec{r}_{ij},h_i) \end{equation} The (magical) value of $N_{ngb}$ is a numerical parameter and its value can be expressed as a function of the more physically relevant parameter $\eta$. In 3D this reads One then change $h$ until an optimal value for $N_{ngb}$ is reached. GADGET uses a bisection algorithm to do so. The (magical) value of $N_{ngb}$ to obtain is a numerical parameter and its value can be expressed as a function of the more physically relevant parameter $\eta$. In 3D the relation between those quantities is \begin{equation} N_{ngb} = \frac{4}{3}\pi\left(\zeta \eta\right)^3 ... ... @@ -212,7 +216,7 @@ and used. \\ Notice that $h$ has to be recomputed through the iterative process presented in the previous section at every time step. The time derivative of the smoothing length only give a rough estimate of its derivative of the smoothing length only gives a rough estimate of its change. It only provides a good guess for the Newton-Raphson (or bisection) scheme. ... ... @@ -221,21 +225,21 @@ bisection) scheme. The usual scheme uses a kick-drift-kick leap-frog integrator. A full time step of size $\Delta t$ consists of the following sub-steps: \\ \textbf{first kick} Compute velocity and internal energy at half step. \textbf{First kick} Compute velocity and internal energy at half step. \begin{eqnarray*} \tilde {\vec{v}}_i &=& \vec{v}_i + \textstyle\frac{1}{2}\Delta t ~\vec{a}_i \\ \tilde u_i &=& u_i + \textstyle\frac{1}{2}\Delta t ~\frac{du_i}{dt} \end{eqnarray*} \textbf{drift} Advance time and position by a full step. \textbf{Drift} Advance time and position by a full step. \begin{eqnarray*} t &\leftarrow& t + \Delta t \\ \vec{x}_i &\leftarrow& \vec{x}_i + \Delta t \tilde {\vec{v}}_i\\ \end{eqnarray*} \textbf{prediction} Estimate velocity, internal energy and smoothing length at full step \textbf{Prediction} Estimate velocity, internal energy and smoothing length at full step \begin{eqnarray*} \vec{v}_i &\leftarrow& \vec{v}_i + \Delta t \vec{a}_i \\ ... ... @@ -243,15 +247,16 @@ u_i &\leftarrow& u_i + \Delta t ~\frac{du_i}{dt} \\ h_i &\leftarrow& h_i + \Delta t ~\frac{dh_i}{dt} \\ \end{eqnarray*} \textbf{SPH loop 1} Compute $\rho_i$, correct $h_i$ and compute $\Omega_i$ using the first SPH loop. \\ \textbf{SPH loop 1} Compute $\rho_i$, correct $h_i$ if needed using bisection algorithm and compute $\Omega_i$ using the first SPH loop. \\ \textbf{SPH loop 2} Compute $\vec{a_i}$ and $\frac{du_i}{dt}$ using the second SPH loop. \\ \textbf{SPH loop 2} Compute $\vec{a_i}$, $\frac{du_i}{dt}$ and $\frac{dh_i}{dt}$ using the second SPH loop. \\ \textbf{Gravity} Compute accelerations due to gravity. \\ \textbf{Cooling} Compute the change in internal energy due to radiative cooling. \\ \textbf{Cooling} Compute the change in internal energy due to radiative cooling and heating. \\ \textbf{second kick} Compute velocity and internal energy at end of step. \textbf{Second kick} Compute velocity and internal energy at end of step. \begin{eqnarray*} \vec{v}_i &=& \tilde{\vec{v}}_i + \textstyle\frac{1}{2}\Delta t ~\vec{a}_i \\ ... ... @@ -272,9 +277,10 @@ E &=&\sum_i m_i\left(\frac{1}{2}|\vec{v_i}|^2+u_i\right)\\ A(s) &=& \left(\gamma -1 \right)\sum_i \frac{u_i}{\rho_i^{\gamma - 1}} \end{eqnarray} The conservation of those quantities in the code depends on the quality of the time integrator. \\ The conservation of those quantities in the code depends on the quality of the time integrator. The leap-frog integrator of the previous section should preserve these quantities to machine precision.\\ Notice that the entropic function $A(s)$ is not the physical'' entropy $s$ but is related to it through a monotonic function. It is just a convenient way to represent entropy. function. It is just a more convenient way to represent entropy. \section{Improved SPH equations} ... ... @@ -340,7 +346,8 @@ using a shock detector and then a slow decay of the viscosity with time. Followi \end{eqnarray} with (usually) $\alpha_{max} = 2$, $\alpha_{min} = 0.1$ and $\sigma=0.1$. The $\alpha$ term in (\ref{eq:visc}) is then replaced by $\bar\alpha = \frac{1}{2}(\alpha_i + \alpha_j)$. The same applies to $\beta$.\\ replaced by $\bar\alpha = \frac{1}{2}(\alpha_i + \alpha_j)$. The same applies to $\beta$ which is now, $\beta = 3\bar\alpha$.\\ The divergence of the velocity field can be computed in the density loop and the exact expression is \begin{equation} ... ... @@ -350,14 +357,15 @@ The divergence of the velocity field can be computed in the density loop and the \subsection{Thermal conductivity} The thermal conductivity dissipating energy at contact discontinuities can be modelled by adding another term to the internal energy equation \ref{eq:dudt}. The thermal conductivity which dissipates energy at discontinuities in the energy field can be modeled by adding another term to the internal energy evolution equation (\ref{eq:dudt}). \begin{equation} \frac{du_i}{dt} \stackrel{cond}{=} - \sum_j \alpha_u v_{sig,u}\left(u_i - u_j\right)\bar{F}_{ij} \end{equation} This time, the signal velocity must vanish when we are dealing with a contact discontinuity. A good choice is to used This time, the signal velocity must vanish when we are dealing with a contact discontinuity as no energy should flow between the two regions in this case. A good choice is to use \begin{equation} v_{sig,u} = \sqrt{\frac{2|P_i-P_j|}{\rho_i+\rho_j}} ... ... @@ -365,12 +373,13 @@ This time, the signal velocity must vanish when we are dealing with a contact di Once again, the $\alpha$-term is made to decay far from any discontinuity. In this case, the equation reads \begin{equation} \frac{d\alpha_{u,i}}{dt} = \frac{h_i \nabla^2 u_i}{10} - \frac{\alpha_{u,i}c_i\sigma}{h_i} \end{equation} \begin{eqnarray} \frac{d\alpha_{u,i}}{dt} &=& \mathcal{S}_u - \frac{\alpha_{u,i}c_i\sigma}{h_i} \\ \mathcal{S}_u &=& \frac{h_i \nabla^2 u_i}{10} \end{eqnarray} where all constants usually take the same value than in the viscosity terms. The laplacian of $u$ can again be computed in the density loop and reads where again $\sigma=0.1$ as in the viscosity terms. As in the velocity divergence case, the laplacian of $u$ can be computed in the density loop and reads \begin{equation} \nabla^2 u_i = \frac{2}{\rho_i} \sum_j m_j \left(u_j - u_i\right) \frac{F_{ij}}{|\vec{r}_{ij}|} ... ...
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