# Particles Unit

The support for particles in Flash-X will eventually be in two flavors, active and passive. This version only supports passive particles. Passive particles acquire their kinematic information (velocities) directly from the mesh. They are meant to be used as passive flow tracers and do not make sense outside of a hydrodynamical context. The governing equation for the $$i$$th passive particle is particularly simple and requires only the time integration of interpolated mesh velocities.

$\label{Eqn:particle_motion_velocity} {d{\bf x}_i\over dt} = {\bf v}_i$

Active particles experience forces and may themselves contribute to the problem dynamics (e.g., through long-range forces or through collisions). They may additionally have their own motion independent of the grid, so an additional motion equation of

$\label{Eq:particle_motion_acceleration} {\bf v}_i^{n+1} = {\bf v}_i^n + {\bf a}_i^n\Delta t^n ~.$

may come into play. Here $${\bf a}_i$$ is the particle acceleration. Solving for the motion of active particles is also referred to as solving the $$N$$-body problem. The equations of motion for the $$i$$th active particle include the equation [Eqn:particle_motion_velocity] and another describing the effects of forces.

$\label{Eqn:particle_motion_forces} m_i{d{\bf v}_i\over dt} = {\bf F}_{{\rm lr,}i} + {\bf F}_{{\rm sr,}i}\ ,$

Here, $${\bf F}_{{\rm lr,}i}$$ represents the sum of all long-range forces (coupling all particles, except possibly those handled by the short-range term) acting on the $$i$$th particle and $${\bf F}_{{\rm sr,}i}$$ represents the sum of all short-range forces (coupling only neighboring particles) acting on the particle.

For both types of particles, the primary challenge is to integrate [Eqn:particle_motion_velocity] forward through time. Many alternative integration methods are described in Section  below. Additional information about the mesh to particle mapping is described in . An introduction to the particle techniques used in Flash-X is given by R. W. Hockney and J. W. Eastwood in Computer Simulation using Particles (Taylor and Francis, 1988).

## Time Integration

The passive particles have many different time integration schemes available. The subroutine Particles/Particles_advance handles the movement of particles through space and time. Because Flash-X will have support for including different types of both active and passive particles in a single simulation, the implementation of Particles/Particles_advance may call several helper routines of the form pt_advanceMETHOD (e.g., pt_advanceLeapfrog, pt_advanceEuler_active, pt_advanceCustom), each acting on an appropriate subset of existing particles. The METHOD here is determined by the ADVMETHOD part of the PARTICLETYPE Config statement (or the ADV par of a -particlemethods option) for the type of particle. See the Particles_advance source code for the mapping from ADVMETHOD keyword to pt_advanceMETHOD subroutine call.

### Passive Particles

Passive particles may be moved using one of several different methods available in Flash-X. With the exception of Midpoint, they are all single-step schemes. The methods are either first-order or second-order accurate, and all are explicit, as described below. In all implementations, particle velocities are obtained by mapping grid-based velocities onto particle positions as described in .

Numerically solving Equation [Eqn:particle_motion_velocity] for passive particles means solving a set of simple ODE initial value problems, separately for each particle, where the velocities $${\bf v}_i$$ are given at certain discrete points in time by the state of the hydrodynamic velocity field at those times. The time step is thus externally given and cannot be arbitrarily chosen by a particle motion ODE solver . Statements about the order of a method in this context should be understood as referring to the same method if it were applied in a hypothetical simulation where evaluations of velocities $${\bf v}_i$$ could be performed at arbitrary times (and with unlimited accuracy). Note that Flash-X does not attempt to provide a particle motion ODE solver of higher accuracy than second order, since it makes little sense to integrate particle motion to a higher order than the fluid dynamics that provide its inputs.

In all cases, particles are advanced in time from $$t^n$$ (or, in the case of Midpoint, from $$t^{n-1}$$) to $$t^{n+1} = t^n + \Delta t^n$$ using one of the difference equations described below. The implementations assume that at the time when Particles/Particles_advance is called, the fluid fields have already been updated to $$t^{n+1}$$, as is the case with the Driver/Driver_evolveFlash implementations provided with Flash-X. A specific implementation of the passive portion of Particles/Particles_advance is selected by a setup option such as -with-unit=Particles/ParticlesMain/passive/Euler, or by specifying something like REQUIRES Particles/ParticlesMain/passive/Euler in a simulation’s Config file (or by listing the path in the Units file if not using the -auto configuration option). Further details are given is below.

• Forward Euler (Particles/ParticlesMain/passive/Euler). Particles are advanced in time from $$t^n$$ to $$t^{n+1} = t^n + \Delta t^n$$ using the following difference equation:

$\label{Eqn:particle_passive_euler} {\bf x}_i^{n+1} = {\bf x}_i^n + {\bf v}_i^n\Delta t^n \quad.$

Here $${\bf v}_i^n$$ is the velocity of the particle, which is obtained using particle-mesh interpolation from the grid at $$t=t^n$$.

Note that this evaluation of $${\bf v}_i^n$$ cannot be deferred until the time when it is needed at $$t=t^{n+1}$$, since at that point the fluid variables have been updated and the velocity fields at $$t=t^n$$ are not available any more. Particle velocities are therefore interpolated from the mesh at $$t=t^n$$ and stored as particle attributes. Similar concerns apply to the remaining methods but will not be explicitly mentioned every time.

• Two-Stage Runge-Kutta (Particles/ParticlesMain/passive/RungeKutta). This 2-stage Runge-Kutta scheme is the preferred choice in Flash-X. It is also the default which is compiled in if particles are included in the setup but no specific alternative implementation is requested. The scheme is also known as Heun’s Method:

\begin{split}\begin{aligned} \label{Eqn:particle_passive_rk2} {\bf x}_i^{n+1} &=& {\bf x}_i^n + \frac{\Delta t^n}{2} \left[ {\bf v}_i^n + {\bf v}_i^{*,n+1} \right] \;,\\ \hbox{where}\quad {\bf v}_i^{*,n+1} &=& {\bf v}({\bf x}_i^{*,n+1},t^{n+1}) \;,\nonumber\\ {\bf x}_i^{*,n+1} &=& {\bf x}_i^n + \Delta t^n {\bf v}_i^n \quad.\nonumber \end{aligned}\end{split}

Here $${\bf v}({\bf x},t)$$ denotes evaluation (interpolation) of the fluid velocity field at position $$\bf x$$ and time $$t$$; $${\bf v}_i^{*,n+1}$$ and $${\bf x}_i^{*,n+1}$$ are intermediate results ; and $${\bf v}_i^n = {\bf v}({\bf x}_i^{n},t^{n})$$ is the velocity of the particle, obtained using particle-mesh interpolation from the grid at $$t=t^n$$ as usual.

• Midpoint (Particles/ParticlesMain/passive/Midpoint). This Midpoint scheme is a two-step scheme. Here, the particles are advanced from time $$t^{n-1}$$ to $$t^{n+1} = t^{n-1} + \Delta t^{n-1} + \Delta t^{n}$$ by the equation

${\bf x}_i^{n+1} = {\bf x}_i^{n-1} + {\bf v}_i^{n}(\Delta t^{n-1} + \Delta t^{n}) \quad.$

The scheme is second order if $$\Delta t^n = \Delta t^{n-1}$$.

To get the scheme started, an Euler step (as described for passive/Euler) is taken the first time Particles/ParticlesMain/passive/Midpoint/pt_advancePassive is called.

The Midpoint alternative implementation uses the following additional particle attributes:

PARTICLEPROP pos2PrevX REAL # two previous x-coordinate PARTICLEPROP pos2PrevY REAL # two previous y-coordinate PARTICLEPROP pos2PrevZ REAL # two previous z-coordinate

• Estimated Midpoint with Correction (Particles/ParticlesMain/passive/EstiMidpoint2). The scheme is second order even if $$\Delta t^n = \Delta t^{n+1}$$ is not assumed. It is essentially the EstiMidpoint or “Predictor-Corrector” method of previous releases, with a correction for non-constant time steps by using additional evaluations (at times and positions that are easily available, without requiring more particle attributes).

Particle advancement follows the equation ‘_=8 ‘_=8

$\label{Eqn:EstiMidpoint2} {\bf x}_i^{n+1} = {\bf x}_i^n + {{\Delta t}}^n \, {\bf v}_i^{\mathrm{comb}}\;,$

where

$\label{Eqn:pt_em2-2} {\bf v}_i^{\mathrm{comb}}= c_1\,{\bf v}({\bf x}_i^{*,n+\frac 12},t^{n}) + c_2\,{\bf v}({\bf x}_i^{*,n+\frac 12},t^{n+1}) + c_3\,{\bf v}({\bf x}_i^{n},t^{n}) + c_4\,{\bf v}({\bf x}_i^{n},t^{n+1})$

is a combination of four evaluations (two each at the previous and the current time),

${\bf x}_i^{*,n+\frac 12}= {\bf x}_i^n + \frac12 \Delta t^{n-1} {\bf v}_i^n$

are estimated midpoint positions as before in the Estimated Midpoint scheme, and the coefficients

\begin{split}\begin{aligned} c_1&=&c_1({\Delta t}^{n-1},{\Delta t}^n)\;,\\ c_2&=&c_2({\Delta t}^{n-1},{\Delta t}^n)\;,\\ c_3&=&c_3({\Delta t}^{n-1},{\Delta t}^n)\;,\\ c_4&=&c_4({\Delta t}^{n-1},{\Delta t}^n)\end{aligned}\end{split}

are chosen dependent on the change in time step so that the method stays second order when $${\Delta t}^{n-1} \neq {\Delta t}^n$$.

Conditions for the correction can be derived as follows: Let $$\Delta t_{*}^{n} =\frac 12 \Delta t^{n-1}$$ the estimated half time step used in the scheme, let $$t_{*}^{n+\frac 12}=t^n + \Delta t_{*}^{n}$$ the estimated midpoint time, and $$t^{n+\frac 12}=t^n + \frac 12 \Delta t^{n}$$ the actual midpoint of the $$[t^n,t^{n+1}]$$ interval. Also write $${\bf x}_i^{{\mathrm{E}},n+\frac 12}= {\bf x}_i^{n}+ \frac 12 \Delta t^{n}{\bf v}_i^{n}$$ for first-order (Euler) approximate positions at the actual midpoint time $$t^{n+\frac 12}$$, and we continue to denote with $${\bf x}_i^{*,n+\frac 12}$$ the estimated positions reached at the estimated mipoint time $$t_{*}^{n+\frac 12}$$.

Assuming reasonably smooth functions $${\bf v}({\bf x},t)$$, we can then write for the exact value of the velocity field at the approximate positions evaluated at the actual midpoint time

$\label{Eqn:vhalf_expansion} {\bf v}({\bf x}_i^{{\mathrm{E}},n+\frac 12},t^{n+\frac 12}) = {\bf v}({\bf x}_i^{n},t^{n}) + {\bf v}_t({\bf x}_i^{n},t^{n})\frac 12 \Delta t^{n}+ ({\bf v}_i^{n}\cdot \frac{\partial}{\partial {\bf x}}) {\bf v}({\bf x}_i^{n},t^{n})\frac 12 \Delta t^{n}+ O((\frac 12 \Delta t^{n})^2)$

by Taylor expansion. It is known that the propagation scheme $$\widetilde{{\bf x}}_i^{n+1}={\bf x}_i^{n}+{\bf v}({\bf x}_i^{{\mathrm{E}},n+\frac 12},t^{n+\frac 12}) {\Delta t}$$ using these velocities is second order (this is known as the modified Euler method).

On the other hand, expansion of [Eqn:pt_em2-2] gives

\begin{split}\begin{aligned} {\bf v}_i^{\mathrm{comb}}&=& (c_1+c_2+c_3+c_4) {\bf v}({\bf x}_i^{n},t^{n}) \nonumber\\ && +\; (c_2+ c_4) {\bf v}_t({\bf x}_i^{n},t^{n}) {\Delta t}\;+\; (c_1+c_2) ({\bf v}_i^{n}\cdot \frac{\partial}{\partial {\bf x}}) {\bf v}({\bf x}_i^{n},t^{n}) \Delta t_{*}^{n} \nonumber\\ && \quad + \quad \mbox{higher order terms in {\Delta t} and \Delta t_{*}^{n} .} \nonumber\end{aligned}\end{split}

After introducing a time step factor $$f$$ defined by $$\Delta t_{*}^{n} = f\,{\Delta t}^{n}$$, this becomes

\begin{split}\begin{aligned} \label{Eqn:vcomb_expansion} {\bf v}_i^{\mathrm{comb}}&=& (c_1+c_2+c_3+c_4) {\bf v}({\bf x}_i^{n},t^{n}) \\ && +\; (c_2+ c_4) {\bf v}_t({\bf x}_i^{n},t^{n}) {\Delta t}\;+\; (c_1+c_2)({\bf v}_i^{n}\cdot \frac{\partial}{\partial {\bf x}}) {\bf v}({\bf x}_i^{n},t^{n}) \,f\,{\Delta t}\nonumber\\ && \quad + \; O(({\Delta t})^2)\quad. \nonumber\end{aligned}\end{split}

One can derive conditions for second order accuracy by comparing [Eqn:vcomb_expansion] with [Eqn:vhalf_expansion] and requiring that

${\bf v}_i^{\mathrm{comb}}= {\bf v}({\bf x}_i^{{\mathrm{E}},n+\frac 12},t^{n+\frac 12}) + O(({\Delta t})^2)\quad.$

It turns out that the coefficients have to satisfy three conditions in order to eliminate from the theoretical difference between numerical and exact solution all $$O({\Delta t}^{n-1})$$ and $$O({\Delta t}^{n})$$ error terms:

\begin{split}\begin{aligned} c_1+c_2+c_3+c_4&=& 1 \quad \mbox{(otherwise the scheme will not even be of first order)}\;,\\ c_2+c_4&=& \frac12 \quad \mbox{(and thus also c_1+c_3=\frac12)}\;, \\ c_1+c_2&=& \frac{ {\Delta t}^n } { {\Delta t}^{n-1} } \quad.\end{aligned}\end{split}

The provided implementation chooses $$c_4=0$$ (this can be easily changed if desired by editing in the code). All four coefficients are then determined:

\begin{split}\begin{aligned} c_1&=&\frac{ {\Delta t}^n } { {\Delta t}^{n-1} } - \frac12\;,\\ c_2&=&\frac12\;,\\ c_3&=&1 - \frac{ {\Delta t}^n } { {\Delta t}^{n-1} }\;,\\ c_4&=&0\quad. \end{aligned}\end{split}

Note that when the time step remains unchanged we have $$c_1=c_2=\frac12$$ and $$c_3=c_4=0$$, and so [Eqn:EstiMidpoint2] simplifies significantly.

An Euler step, as described for passive/Euler in [Eqn:particle_passive_euler], is taken the first time when Particles/ParticlesMain/passive/EstiMidpoint2/pt_advancePassive is called and when the time step has changed too much. Since the integration scheme is tolerant of time step changes, it should usually not be necessary to apply the second criterion; even when it is to be employed, the criteria should be less strict than for an uncorrected EstiMidpoint scheme. For EstiMidPoint2 the timestep is considered to have changed too much if either of the following is true:

$\Delta t^{n} > \Delta t^{n-1} \quad\mbox{and}\quad \left| \Delta t^{n} - \Delta t^{n-1} \right| \geq {\tt pt\_dtChangeToleranceUp} \times \Delta t^{n-1}$

or

$\Delta t^{n} < \Delta t^{n-1} \quad\mbox{and}\quad \left| \Delta t^{n} - \Delta t^{n-1} \right| \geq {\tt pt\_dtChangeToleranceDown} \times \Delta t^{n-1},$

where Particles/pt_dtChangeToleranceUp and Particles/pt_dtChangeToleranceDown are runtime parameter specific to the EstiMidPoint2 alternative implementation.

The EstiMidpoint2 alternative implementation uses the following additional particle attributes for storing the values of $${\bf x}_i^{*,n+\frac 12}$$ and $${\bf v}_i^{*,n+\frac 12}$$ between the Particles_advance calls at $$t^n$$ and $$t^{n+1}$$:

PARTICLEPROP velPredX REAL PARTICLEPROP velPredY REAL PARTICLEPROP velPredZ REAL PARTICLEPROP posPredX REAL PARTICLEPROP posPredY REAL PARTICLEPROP posPredZ REAL

The time integration of passive particles is tested in the ParticlesAdvance unit test, which can be used to examine the convergence behavior, see .

## Mesh/Particle Mapping

Particles behave in a fundamentally different way than grid-based quantities. Lagrangian, or passive particles are essentially independent of the grid mesh and move along with the velocity field. Active particles may be located independently of mesh refinement. In either case, there is a need to convert grid-based quantities into similar attributes defined on particles, or vice versa. The method for interpolating mesh quantities to tracer particle positions must be consistent with the numerical method to avoid introducing systematic error. In the case of a finite-volume methods such as those used in Flash-X, the mesh quantities have cell-averaged rather than point values, which requires that the interpolation function for the particles also represent cell-averaged values. Cell averaged quantities are defined as

$f_i\left(x\right) \: \equiv \: \frac{1}{\Delta x} \int_{x_{i-1/2}}^{x_{i-1/2}} f\left(x^\prime \right) dx^\prime$

where $$i$$ is the cell index and $$\Delta x$$ is the spatial resolution. The mapping back and forth from the mesh to the particle properties are defined in the routines Particles_mapFromMesh and Particles_mapToMeshOneBlk.

Specifying the desired mapping method is accomplished by designating the MAPMETHOD in the Simulation Config file for each type of particle. See for more details.

The quadratic mapping package defines an interpolation back and forth to the mesh which is second order. This implementation is primarily meant to be used with passive tracer particles.

To derive it, first consider a second-order interpolation function of the form

$f\left(x\right) \: = \: A + B \left(x - x_i\right) + C \left(x - x_i\right)^2 \; .$

Then integrating gives

\begin{split}\begin{aligned} f_{i-1} \: & = & \: \frac{1}{\Delta x}\left[ A + \left. \frac{1}{2} B \left(x - x_i\right)^2 \right|^{x_{i-1/2}}_{x_{i-3/2}} + \left. \frac{1}{3} C \left(x - x_i\right)^3 \right|^{x_{i-1/2}}_{x_{i-3/2}} \right]\nonumber \\ & = & \: A - B\Delta x + \frac{13}{12} C \Delta x^2 \;,\end{aligned}\end{split}
\begin{split}\begin{aligned} f_{i} \: & = & \: \frac{1}{\Delta x}\left[ A + \left. \frac{1}{2} B \left(x - x_i\right)^2 \right|^{x_{i+1/2}}_{x_{i-1/2}} + \left. \frac{1}{3} C \left(x - x_i\right)^3 \right|^{x_{i+1/2}}_{x_{i-1/2}} \right]\nonumber \\ & = & \: A + \frac{1}{12} C \Delta x^2 \; ,\end{aligned}\end{split}

and

\begin{split}\begin{aligned} f_{i+1} \: & = & \: \frac{1}{\Delta x}\left[ A + \left.\frac{1}{2} B \left(x - x_i\right)^2 \right|^{x_{i+3/2}}_{x_{i+1/2}}. + \left. \frac{1}{3} C \left(x - x_i\right)^3 \right|^{x_{i-1/2}}_{x_{i-3/2}} \right]\nonumber \\ & = & \: A - B\Delta x + \frac{13}{12} C \Delta x^2 \;,\end{aligned}\end{split}

We may write these as

$\begin{split}\left[ \begin{array}{c} f_{i+1} \\ f_{i} \\ f_{i-1} \end{array} \right] \: = \: \left[ \begin{array}{ccc} 1 & -1 & \frac{13}{12} \\ 1 & 0 & \frac{1}{12} \\ 1 & 1 & \frac{13}{12} \end{array} \right] \left[ \begin{array}{ccc} A \\ B\Delta x \\ C \Delta x^2 \end{array} \right] \; .\end{split}$

Inverting this gives expressions for $$A$$, $$B$$, and $$C$$,

$\begin{split}\left[ \begin{array}{c} A \\ B\Delta x \\ C \Delta x^2 \end{array} \right] \: = \: \left[ \begin{array}{ccc} -\frac{1}{24} & \frac{13}{12} & -\frac{1}{24} \\ -\frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & -1 & \frac{1}{2} \end{array} \right] \left[ \begin{array}{ccc} f_{i+1} \\ f_{i} \\ f_{i-1} \end{array} \right] \; .\end{split}$

In two dimensions, we want a second-order interpolation function of the form

$f\left(x,y\right) \: = \: A + B \left(x - x_i\right) + C \left(x - x_i\right)^2 + D \left(y - y_j\right) + E \left(y - y_j\right)^2 + F \left(x - x_i\right)\left(y - y_j\right) \; .$

In this case, the cell averaged quantities are given by

$f_{i,j}\left(x,y\right) \: \equiv \: \frac{1}{\Delta y}{\Delta x} \int_{x_{i-1/2}}^{x_{i+1/2}} dx^\prime \int_{y_{j-1/2}}^{x_{j-1/2}} dy^\prime f\left(x^\prime,y^\prime \right) \; .$

Integrating the 9 possible cell averages gives, after some algebra,

$\begin{split}\left[ \begin{array}{c} f_{i-1,j-1} \\ f_{i,j-1} \\ f_{i+1,j-1} \\ f_{i-1,j} \\ f_{i,j} \\ f_{i+1,j} \\ f_{i-1,j+1} \\ f_{i,j+1} \\ f_{i+1,j+1} \end{array} \right] \: = \: \left[ \begin{array}{cccccc} 1 & -1 & \frac{13}{12} & -1 & \frac{13}{12} & 1 \\ 1 & 0 & \frac{1}{12} & -1 & \frac{13}{12} & 0 \\ 1 & 1 & \frac{13}{12} & -1 & \frac{13}{12} & -1 \\ 1 & -1 & \frac{13}{12} & 0 & \frac{1}{12} & 0 \\ 1 & 0 & \frac{1}{12} & 0 & \frac{1}{12} & 0 \\ 1 & 1 & \frac{13}{12} & 0 & \frac{1}{12} & 0 \\ 1 & -1 & \frac{13}{12} & 1 & \frac{13}{12} & -1 \\ 1 & 0 & \frac{1}{12} & 1 & \frac{13}{12} & 0 \\ 1 & 1 & \frac{13}{12} & 1 & \frac{13}{12} & 1 \end{array} \right] \left[ \begin{array}{c} A \\ B\Delta x \\ C \Delta x^2 \\ D \Delta y \\ E \Delta y^2 \\ F \Delta x \Delta y \end{array} \right] \; .\end{split}$

At this point we note that there are more constraints than unknowns, and we must make a choice of the constraints. We chose to ignore the cross terms and take only the face-centered cells next to the cell containing the particle, giving

$\begin{split}\left[ \begin{array}{c} f_{i,j-1} \\ f_{i-1,j} \\ f_{i,j} \\ f_{i+1,j} \\ f_{i,j+1} \end{array} \right] \: = \: \left[ \begin{array}{cccccc} 1 & 0 & \frac{1}{12} & -1 & \frac{13}{12} \\ 1 & -1 & \frac{13}{12} & 0 & \frac{1}{12} \\ 1 & 0 & \frac{1}{12} & 0 & \frac{1}{12} \\ 1 & 1 & \frac{13}{12} & 0 & \frac{1}{12} \\ 1 & 0 & \frac{1}{12} & 1 & \frac{13}{12} \end{array} \right] \left[ \begin{array}{c} A \\ B\Delta x \\ C \Delta x^2 \\ D \Delta y \\ E \Delta y^2 \end{array} \right] \; .\end{split}$

Inverting gives

$\begin{split}\left[ \begin{array}{c} A \\ B\Delta x \\ C \Delta x^2 \\ D \Delta y \\ E \Delta y^2 \end{array} \right] \: = \: \left[ \begin{array}{cccccc} -\frac{1}{24} & -\frac{1}{24} & \frac{7}{6} & -\frac{1}{24} & -\frac{1}{24} \\ 0 & -\frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & -1 & \frac{1}{2} & 0 \\ -\frac{1}{2} & 0 & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & -1 & 0 & \frac{1}{2} \end{array} \right] \left[ \begin{array}{c} f_{i,j-1} \\ f_{i-1,j} \\ f_{i,j} \\ f_{i+1,j} \\ f_{i,j+1} \end{array} \right] \; .\end{split}$

Similarly, in three dimensions, the interpolation function is

$f\left(x,y,z\right) \: = \: A + B \left(x - x_i\right) + C \left(x - x_i\right)^2 + D \left(y - y_j\right) + E \left(y - y_j\right)^2 + F \left(z - z_k\right) + G \left(z - z_k\right)^2 \; .$

and we have

$\begin{split}\left[ \begin{array}{c} A \\ B \Delta x \\ C {\Delta x}^2 \\ D \Delta y \\ E {\Delta y}^2 \\ F \Delta z \\ G {\Delta z}^2 \end{array} \right] \: = \: \left[ \begin{array}{ccccccc} -\frac{1}{24} & -\frac{1}{24} & -\frac{1}{24} & \frac{5}{4} & -\frac{1}{24} & -\frac{1}{24} & -\frac{1}{24} \\ 0 & 0 & -\frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{2} & -1 & \frac{1}{2} & 0 & 0 \\ 0 & -\frac{1}{2} & 0 & 0 & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & -1 & 0 & \frac{1}{2} & 0 \\ -\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & 0 & -1 & 0 & 0 & \frac{1}{2} \end{array} \right] \left[ \begin{array}{c} f_{i,j,k-1} \\ f_{i,j-1,k} \\ f_{-i,j,k} \\ f_{i,j,k} \\ f_{i+1,j,k} \\ f_{i,j+1,k} \\ f_{i,j,k+1} \end{array} \right] \; .\end{split}$

Finally, the above expressions apply only to Cartesian coordinates. In the case of cylindrical $$\left(r,z\right)$$ coordinates, we have

\begin{split}\begin{aligned} f\left(r,z\right) \; = & & \nonumber \\ & A + B \left(r - r_i\right) + C \left(r - r_i\right)^2 + D \left(z - z_j\right) & \nonumber \\ & + E \left(z - z_j\right)^2 + F \left(r - r_i\right)\left(z - z_j\right) \; . &\end{aligned}\end{split}

and

\begin{split}\begin{aligned} \left[ \begin{array}{c} A \\ B\Delta r \\ C\Delta r^{\frac{2}{6}} \\ D\Delta z \\ E \Delta z^2 \end{array} \right] \: = \hspace{-1.0in} & & \nonumber \\ & \left[ \begin{array}{cccccc} -\frac{1}{24} & -\frac{h_1-1}{24h_1} & \frac{7}{6} & -\frac{h_1-1}{24h1} & -\frac{1}{24} \\ 0 & - \frac{\left(7+6h_1\right)\left(h_1-1\right)}{3h_2} & \frac{2h_1}{3h2} & \frac{\left(7+6h_1\right)\left(h_1-1\right)}{3h_2} & 0 \\ 0 & \frac{\left(12h_1^2 + 12 h_1 - 1\right)\left(h_1 - 1\right)}{h_1h_2} & -2 \frac{12 h_1^2 -13}{h_2} & - \frac{\left(12h_1^2 + 12 h_1 - 1\right)\left(h_1 - 1\right)}{h_1h_2} & 0 \\ - \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & - 1 & \frac{1}{2} & 0 \\ \end{array} \right] \left[ \begin{array}{c} f_{i,j-1} \\ f_{i-1,j} \\ f_{i,j} \\ f_{i+1,j} \\ f_{i,j+1} \end{array} \right] \; . &\end{aligned}\end{split}

### Cloud in Cell Mapping

Other interpolation routines can be defined that take into account the actual quantities defined on the grid. These “mesh-based” algorithms are represented in Flash-X by the Cloud-in-Cell mapping, where the interpolation to/from the particles is defined as a simple linear weighting from nearby grid points. The weights are defined by considering only the region of one “cell” size around each particle location; the proportional volume of the particle “cloud” corresponds to the amount allocated to/from the mesh. The CIC method can be used with both types of particles. When using it with active particles the MapToMesh methods should also be selected. In order to include the CIC method with passive particles, the setup command line option is -with-unit=Particles/ParticlesMapping/CIC. Two additional command line option -with-unit=Particles/ParticlesMapping/MapToMesh and -with-unit=Grid/GridParticles/MapToMesh are necessary when using the active particles. All of these command line options can be replaced by placing the appropriate REQUIRES/REQUESTS directives in the Simulation Config file.

## Using the Particles Unit

The Particles unit encompasses nearly all aspects of Lagrangian particles. The exceptions are input/output the movement of related data structures between different blocks as the particles move from one block to another, and mapping the particle attributes to and from the grid.

Particle types must be specified in the Config file of the Simulations unit setup directory for the application, and the syntax is explained in . At configuration time, the setup script parses the PARTICLETYPE specifications in the Config files, and generates an F90 file Particles/Particles_specifyMethods.F90 that populates a data structure gr_ptTypeInfo. This data structure contains information about the method of initialization and interpolation methods for mapping the particle attributes to and from the grid for each included particle type. Different time integration schemes are applied to active and passive particles. However, in one simulation, all active particles are integrated using the same scheme, regardless of how many active types exists. Similarly, only one passive integration scheme is used. The added complexity of multiple particle types allows different methods to be used for initialization of particles positions and their mapping to and from the grid quantities. Because several different implementations of each type of functionality can co-exist in one simulation, there are no defaults in the Particles unit Config files. These various functionalities are organized into different subunits; a brief description of each subunit is included below and further expanded in subsections in this chapter.

• The ParticlesInitialization subunit distributes a given set of particles through the spatial domain at the simulation startup. Some type of spatial initialization is always required; the functionality is provided by Particles/Particles_initPositions. The users of active particles typically have their own custom initialization. The following two implementations of initialization techniques are included in the Flash-X distribution (they are more likely to used with the passive tracer particles):

Lattice

distributes particles regularly along the axes directions throughout a subsection of the physical grid.

WithDensity

distributes particles randomly, with particle density being proportional to the grid gas density.

Users have two options for implementing custom initialization methods. The two files involved in the process are: Particles/Particles_initPositions and pt_initPositions. The former does some housekeeping such as allowing for inclusion of one of the available methods along with the user specified one, and assigning tags at the end. A user wishing to add one custom method with no constraints on tags etc is advised to implement a custom version of the latter. This approach allows the user to focus the implementation on the placement of particles only. Users desirous of refining the grid based on particles count during initialiation should see the setup PoisParticles for an example implementation of the Particles_initPositions routine. If more than one implementation of pt_initPositions is desired in the same simulation then it is necessary to implement each one separately with different names (as we do for tracer particles: pt_initPositionsLattice and pt_initPositionsWithDensity) in their simulation setup directory. In addition, a modified copy of Particles_initPostions, which calls these new routines in the loop over types, must also be placed in the same directory.

• The ParticlesMain subunit contains the various time-integration options for both active and passive particles. A detailed overview of the different schemes is given in .

• The ParticlesMapping subunit controls the mapping of particle properties to and from the grid. Flash-X currently supplies the following mapping schemes:

Cloud-in-cell

(ParticlesMapping/meshWeighting/CIC), which weights values at nearby grid cells; and

Quadratic

(ParticlesMapping/Quadratic), which performs quadratic interpolation.

Some form of mapping must always be included when running a simulation with particles. As mentioned in the quadratic mapping scheme is only available to map from the grid quantities to the corresponding particle attributes. Since active particles require the same mapping scheme to be used in mapping to and from the mesh, they cannot use the quadratic mapping scheme as currently implemented in Flash-X. The CIC scheme may be used by both the active and passive particles.

After particles are moved during time integration or by forces, they may end up on other blocks within or without the current processor. The redistribution of particles among processors is handled by the GridParticles subunit, as the algorithms required vary considerably between the grid implementations. The boundary conditions are also implemented by the GridParticles unit. See for more details of these redistribution algorithms. The user should include the option -with-unit=Grid/GridParticles on the setup line, or REQUIRES Grid/GridParticles in the Config file.

In addition, the input-output routines for the Particles unit are contained in a subunit IOParticles. Particles are written to the main checkpoint files. If the user desires, a separate output file can be created which contains only the particle information. See below as well as for more details. The user should include the option -with-unit=IO/IOParticles on the setup line, or REQUIRES IO/IOParticles in the Config file.

In Flash-X, the initial particle positions can be used to construct an appropriately refined grid, i.e. more refined in places where there is a clustering of particles. To use this feature the flash.par file must include: refine_on_particle_count=.true. and max_particles_per_blk=[some value]. Please be aware that Flash-X will abort if the criterion is too demanding. To overcome the abort, specify a less demanding criterion, or increase the value of lrefine_max.

### Particles Runtime Parameters

There are several general runtime parameters applicable to the Particles unit, which affect every implementation. The variable Particles/useParticles obviously must be set equal to .true. to utilize the Particles unit. The time stepping is controlled with Particles/pt_dtFactor; a value less than one ensures that particles will not step farther than one entire cell in any given time interval. The Lattice initialization routines have additional parameters. The number of evenly spaced particles is controlled in each direction by Particles/pt_numX and similar variables in $$Y$$ and $$Z$$. The physical range of initialization is controlled by Particles/pt_initialXMin and the like. Finally, note that the output of particle properties to special particle files is controlled by runtime parameters found in the IO unit. See for more details.

### Particle Attributes

By default, particles are defined to have eight real properties or attributes: 3 positions in x,y,z; 3 velocities in x,y,z; the current block identification number; and a tag which uniquely identifies the particle. Additional properties can be defined for each particle. For example, active particles usually have the additional properties of mass and acceleration (needed for the integration routines, see Table ). Depending upon the simulation, the user can define particle properties in a manner similar to that used for mesh-based solution variables. To define a particle attribute, add to a Config file a line of the form

PARTICLEPROP property-name

For attributes that are meant to merely sample and record the state of certain mesh variables along trajectories, Flash-X can automatically invoke interpolation (or, in general, some map method) to generate attribute values from the appropriate grid quantities. (For passive tracer particles, these are typically the only attributes beyond the default set of eight mentioned above.) The routine Particles/Particles_updateAttributes is invoked by Flash-X at appropriate times to effect this mapping, namely before writing particle data to checkpoint and particle plot files. To direct the default implementation of Particles_updateAttributes to act as desired for tracer attributes, the user must define the association of the particle attribute with the appropriate mesh variable by including the following line in the Config file:

PARTICLEMAP TO property-name FROM VARIABLE variable-name

These particle attributes are carried along in the simulation and output in the checkpoint files. At runtime, the user can specify the attributes to output through runtime parameters Particles/particle_attribute_1, Particles/particle_attribute_2, etc. These specified attributes are collected in an array by the Particles/Particles_init routine. This array in turn is used by Particles/Particles_updateAttributes to calculate the values of the specified attributes from the corresponding mesh quantities before they are output.

### Particle I/O

Particle data are written to and read from checkpoint files by the I/O modules (). For more information on the format of particle data written to output files, see and .

Particle data can also be written out to the flash.dat file. The user should include a local copy of IO/IO_writeIntegralQuantities in their Simulation directory. The Orbit test problem supplies an example IO_writeIntegralQuantities routine that is useful for writing individual particle trajectories to disk at every timestep.

There is also a utility routine Particles/Particles_dump which can be used to dump particle output to a plain text file. An example of usage can be found in Particles/Particles_unitTest. Output from this routine can be read using the fidlr routine particles_dump.pro.