### Updated the theory file with a more readable version of the integration scheme...

Updated the theory file with a more readable version of the integration scheme and multiple time step scheme.

Former-commit-id: 4568b597e7d303614d82f6db879460ea4909c1ba
parent 1b25dea6
 ... ... @@ -227,47 +227,101 @@ bisection) scheme. \section{Time integration} 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: \\ The usual scheme uses a kick-drift-kick leap-frog integrator. The various sub-steps are: \textbf{First kick} Compute velocity and internal energy at half step. \textbf{First kick} Kick the particle for the first 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} v_i^{n+1/2} &=& v_i^{n} + \frac{1}{2} a_i^{n}\Delta t^n \\ u_i^{n+1/2} &=& u_i^{n} + \frac{1}{2} \frac{du_i^n}{dt}\Delta t^n \\ \end{eqnarray*} \textbf{Drift} Advance time and position by a full step. \textbf{Particle drift} Advance particles 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*} \begin{equation*} x_i^{n+1} = x_i^n + v_i^{n+{1/2}} \Delta t^n \end{equation*} \textbf{Prediction} Estimate velocity, internal energy and smoothing length at full step \textbf{Prediction} Predict particles forward in time according to their old accelerations'' \begin{eqnarray*} \vec{v}_i &\leftarrow& \vec{v}_i + \Delta t \vec{a}_i \\ u_i &\leftarrow& u_i + \Delta t ~\frac{du_i}{dt} \\ h_i &\leftarrow& h_i + \Delta t ~\frac{dh_i}{dt} \\ v_{i,pred} &=& v_{i,pred} + a_i^{n} \Delta t^n \\ u_{i,pred} &=& u_{i,pred} \cdot \exp\left(\frac{1}{u_i^n}\frac{du_i^{n}}{dt} \Delta t^n\right)\\ \rho_i^{n+1} &=& \rho_i^n \cdot \exp\left(\frac{-3}{h_i^n} \frac{dh_i^{n}}{dt} \Delta t^n \right) \\ h_i^{n+1} &=& h_i^n \cdot \exp\left(\frac{1}{h_i^n} \frac{dh_i^{n}}{dt} \Delta t^n \right) \\ \Omega_i^{n+1} &=& \Omega_i \end{eqnarray*} \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 1} Correct $\rho_i^{n+1}$, $h_i^{n+1}$ and $\Omega_i^{n+1}$ if needed using bisection algorithm and the first SPH loop (Section \ref{sec:density}). \\ \textbf{SPH loop 2} Compute $a_i^{n+1}$, $\frac{du_i^{n+1}}{dt}$ and $\frac{dh_i^{n+1}}{dt}$ using the second SPH loop (Section \ref{sec:forces}). This loop is ALWAYS performed using the PREDICTED velocity $v_{i,pred}$ and internal energy $u_{i,pred}$ for BOTH particles. \\ \textbf{New time step} Calculate the next time step. \begin{equation*} \Delta t^{n+1} = C_{CFL} \frac{h_i}{c_i}, \qquad c_i =\sqrt{\gamma (\gamma-1)u_{i,pred}} \end{equation*} \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 and heating. \\ \textbf{Second kick} Compute velocity and internal energy at end of step. \textbf{Second kick} Kick the particle for the second half-step. \begin{eqnarray*} \vec{v}_i &=& \tilde{\vec{v}}_i + \textstyle\frac{1}{2}\Delta t ~\vec{a}_i \\ u_i &=& \tilde{u}_i + \textstyle\frac{1}{2}\Delta t ~\frac{du_i}{dt} v_i^{n+1} &=& v_i^{n+1/2} + \frac{1}{2} a_i^{\textcolor{red}{n+1}}\Delta t^{\textcolor{red}{n}} \\ u_i^{n+1} &=& u_i^{n+1/2} + \frac{1}{2} \frac{du_i^{\textcolor{red}{n+1}}}{dt}\Delta t^{\textcolor{red}{n}} \\ v_{i,pred} &=& v_i^{n+1} \\ u_{i,pred} &=& u_i^{n+1} \\ \end{eqnarray*} % 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. % % \begin{eqnarray*} % \vec{v}_i^{n+1/2} &=& \vec{v}_i^n + \textstyle\frac{1}{2}\Delta t^n ~\vec{a}_i\big|^{n-1/2} \\ % u_i^{n+1/2} &=& u_i^n + \textstyle\frac{1}{2}\Delta t^n ~\frac{du_i}{dt}\big|^{n-1/2} % \end{eqnarray*} % % \textbf{Drift} Advance time and position by a full step. % % \begin{eqnarray*} % t &\leftarrow& t + \Delta t^n \\ % \vec{x}_i^{n+1} &\leftarrow& \vec{x}_i^{n+1} + \Delta t^n \tilde {\vec{v}}_i^{n}\\ % \end{eqnarray*} % % \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 \\ % 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 $h_i$ if needed using bisection algorithm and compute $\Omega_i$ using the % first 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 and heating. \\ % % \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 \\ % u_i &=& \tilde{u}_i + \textstyle\frac{1}{2}\Delta t ~\frac{du_i}{dt} % \end{eqnarray*} \section{Multiple time steps} In most of the astrophysical applications, the range of time steps of the different particles is huge. In order to ... ... @@ -300,24 +354,10 @@ new bin is active. In other words, particles can always move down the time bin h two bins of interest are synchronized.\\ When an active particles is integrated in time, it will probably have to interact with inactive neighbors. The state of these inactive particles has to be \emph{predicted} forward in time from the last time they have been active. To do so, the following approximate evolution equations are used: \begin{eqnarray} \vec{x}_i &\leftarrow& \vec{x}_{i} + \vec{v}_i \cdot \Delta t \\ \vec{v}_{i,pred} &\leftarrow& \vec{v}_{i,pred} + \vec{a}_i \cdot \Delta t \\ \rho_i &\leftarrow& \rho_i \cdot \exp\left(\frac{-3}{h_i} \frac{dh_i}{dt} \Delta t \right) \\ h_i &\leftarrow& h_i \cdot \exp\left(\frac{1}{h_i} \frac{dh_i}{dt} \Delta t \right) \\ u_{i,pred} &\leftarrow& u_{i,pred} \cdot \exp\left(\frac{1}{u_i}\frac{du_i}{dt} \Delta t\right) \\ \Omega_i &\leftarrow& 1 \end{eqnarray} These predicted quantities are then used in equations \ref{eq:pressure}, \ref{eq:acceleration} and \ref{eq:dudt} to estimate the acceleration and change in internal energy of an active particle. In order to preserve the old value of the velocity and internal energy for accurate time integration, we need to add 2 additional variables to the particles, the predicted velocity $\vec{v}_{i,pred}$(vector) and the predicted internal energy $u_{i,pred}$(scalar). The true velocity $\vec{v}_i$ and true internal energy $u_i$ are NOT updated when predicting the state of a particle at an intermediate time step. of these inactive particles has to be predicted forward in time from the last time they have been active. To do so, the prediction part of the leapfrog algorithm is used. \\ In other words, the drift and prediction parts of the leapfrog algorithm are performed for all particles whereas the other parts are only done for the active ones. \section{Conserved quantities} ... ...
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