§3.8 无网格方法

A new class of Lagrangian methods, the so-called meshless formulations, have been recently developed for astrophysical problems. More details can be found in ^[102] ,^[104] ,^[105] , which follow earlier, pioneering work by ^[106] ,^[107] ,^[108] . In short, the derivation starts from the integral form $$ \int[u(\vec{x},t)\dot\phi(\vec{x},t) + \vec{F}(u,\vec{x},t)\cdot\nabla\phi(\vec{x},t) + S(\vec{x},t)\phi(\vec{x},t)]\,d\vec{x}\,dt = 0 $$ of a scalar conservation law $$ \frac{\partial u}{\partial t} + \nabla\cdot(\vec{F} + \vec{a}u) = S \,. $$ Here, $u(\vec{x},t)$ is a scalar field, $S(\vec{x}, t)$ is its source, $\vec{F}(u,\vec{x},t)$ is its flux in a frame moving with velocity $\vec{a}(\vec{x}, t)$, and $\phi(\vec{x},t)$ is an arbitrary differentiable function in space and time leading to the advective derivative $\dot\phi(\vec{x}, t) = \partial\phi(\vec{x}, t)/\partial t + \vec{a}(x,t)\cdot\nabla\phi(\vec{x}, t)$.

Using a discretization through a set of particles $i$ with a smoothing length $h(\vec{x})$ and a kernel function $W(\vec{x},h)$, akin to what is done in SPH, the partitioning of the particles can be written as $$ \psi_i(\vec{x}) = w(\vec{x})W(\vec{x}-\vec{x}_i,h(\vec{x})), $$ where the number density of particles is $w(\vec{x})^{-1} = \sum_j W(\vec{x}-\vec{x}_j,h(\vec{x}))$. The discretization of an arbitrary function $f(\vec{x})$ can be written as $$ \int f(\vec{x})\,d\vec{x} \approx \sum_i f_i \int \psi_i(\vec{x})\,d\vec{x} \equiv \sum_i f_i V_i, $$ where $V_i = \int \psi_i(\vec{x})\,d\vec{x}$ is the effective volume of a particle $i$. Therefore, the discrete form of the integral equation (3.28) can be written as $$ \sum_i \int [ V_i u_i \dot\phi_i + V_i F^\alpha_i (D^\alpha\varphi)_i + V_i S_i \phi_i ] = 0 \, . $$ Although in principle an SPH estimate for $(D^\alpha\phi)_i$ could be used, it is convenient to use a more accurate meshless gradient estimate, as suggested by ^[107] . This, along with the integration of the first term by parts, makes it possible (as shown in ^[102] ) to split $\phi(\vec{x}, t)$ and to obtain: $$ \frac{d}{dt}(V_i u_i) + \sum_j\left[V_i F_i^\alpha\psi_j^\alpha(\vec{x}_i) - V_j F_j^\alpha\psi_i^\alpha(\vec{x}_j)\right] = V_i S_i \, . $$ This equation (and its extension to a general vector field $\vec{u}$) is very similar to the finite volume equation for the moving mesh method [Fn: Note that, however, Eq. [eq:flux] is in the integral form.] but also somewhat similar to the SPH equations, except that the interactions between different particles is described in the source and flux terms, which can be obtained as the solution of an approximate Riemann problem between particles $i$ and $j$. A closer inspection also reveals that only the projection on the direction between the particles is needed, e.g., the solution of the Riemann problem at the midpoint\footnote{More accurate would be the quadrature point at an equal fraction of the kernel length $h_i$ and $h_j$, see discussion in ^[104] and references therein.}. This also means that the primitive variables have to be extrapolated to the midpoint, e.g., using linear extrapolation, and generally a flux limiter has to be applied (see discussion in ^[102] ,^[104] ,^[105] ). Fig. 7 illustrates the differences in partitioning the volumes between mesh-less methods, unstructured (moving) grid and classical kernel weighted formalism (SPH).