update

docs/src/details.md

@@ -1,8 +1,53 @@
 ## Details
 
-`m_i, m_j`
+### Denotation
 
-`m_0`
+Particle indexes: ``i``, ``j``;
+
+``W`` - kernel function value;
+
+``\nabla W`` - kernel function gradient;
+
+``h`` -  smoothing length;
+
+``H ~ 2h`` - kernel support length;
+
+``m, m_i, m_j``  - mass;
+
+``m_0`` - reference mass;
+
+``\rho. \rho_i, \rho_j`` - dencity;
+
+``\rho_0`` - reference dencity;
+
+``p, p_i, p_j`` - pressure;
+
+``\Pi`` - artificial viscosity term;
+
+``\textbf{r}, \textbf{r}_i, \textbf{r}_j`` - particle location (vector);
+
+``\textbf{r}_{ij} = \textbf{r}_{i} - \textbf{r}_{j}`` - particle ``i`` to ``j`` distance (vector);
+
+``r_{ij} = \sqrt{\textbf{r}_{ji} \cdot \textbf{r}_{ji}}`` - particle ``i`` to ``j`` distance;
+
+``\textbf{v}, \textbf{v}_i, \textbf{v}_j`` - velocity (vector);
+
+``\textbf{v}_{ij} = \textbf{v}_{i} - \textbf{v}_{j}`` - relative velocity  (vector);
+
+``V, V_i, V_j`` - volume;
+
+``V_i = \frac{m_i}{\rho_i}; V_j = \frac{m_j}{\rho_j}``
+
+
+### Constants
+
+``g`` - gravity;
+
+``c₀`` - speed of sound;
+
+``γ = 7`` - gamma costant (pressure equation of state);
+
+``\delta_{\Phi}`` - coefficient for density diffusion;
 
 
 
@@ -15,10 +60,12 @@
 
 \\
 \\
+
 \overline{c}_{ij}  = \frac{c_i + c_j}{2}
 
 \\
 \\
+
 \overline{\rho}_{ij} = \frac{\rho_i + \rho_j}{2}
 
 ```
@@ -28,6 +75,7 @@ Monaghan style artificial viscosity:
 ```math
 
 \frac{\partial \textbf{v}_i}{\partial t} = - \sum  m_j \Pi_{ij} \nabla_i W_{ij}
+
 ```
 
 J. Monaghan, “Smoothed particle hydrodynamics”, Reports on Progress in Physics, 68 (2005), pp. 1703-1759.
@@ -36,7 +84,7 @@ J. Monaghan, “Smoothed particle hydrodynamics”, Reports on Progress in Physi
 ### Momentum Equation with Artificial Viscosity
 
 ```math
-\frac{\partial \textbf{v}_i}{\partial t} = - \sum  m_j \left( \frac{b_i}{\rho^2_i} + \frac{b_j}{\rho^2_j} + \Pi_{ij} \right) \nabla_i W_{ij}
+\frac{\partial \textbf{v}_i}{\partial t} = - \sum  m_j \left( \frac{p_i}{\rho^2_i} + \frac{p_j}{\rho^2_j} + \Pi_{ij} \right) \nabla_i W_{ij}
 
 ```
 
@@ -51,6 +99,12 @@ J. Monaghan, Smoothed Particle Hydrodynamics, “Annual Review of Astronomy and
 \frac{\partial \rho_i}{\partial t} = \sum  m_j \textbf{v}_{ij} \cdot \nabla_i W_{ij} + \delta_{\Phi} h c_0 \sum \Psi_{ij} \cdot \nabla_i W_{ij} \frac{m_j}{\rho_j}
 
 
+\\
+\\
+
+
+\Psi_{ij} = 2 (\rho_{ij}^T + \rho_{ij}^H) \frac{\textbf{r}_{ij}}{r_{ij}^2 + \eta^2}
+
 ```
 
 
@@ -67,24 +121,30 @@ J. Monaghan, Smoothed Particle Hydrodynamics, “Annual Review of Astronomy and
 
 
 ```math
+
 \delta \textbf{v}_i^{DPC} = \sum k_{ij}\frac{m_j}{m_i + m_j}v_{ij}^{coll} + \frac{\Delta  t}{\rho_i}\sum \phi_{ij} \frac{2V_j}{V_i + V_j}\frac{p_{ij}^b}{r_{ij}^2 + \eta^2}\textbf{r}_{ij}
 
+\\
 \\
 
 (v_{ij}^{coll} , \quad \phi_{ij}) = \begin{cases} (\frac{\textbf{v}_{ij}\cdot \textbf{r}_{ij}}{r_{ij}^2 + \eta^2}\textbf{r}_{ji}, \quad 0) & \textbf{v}_{ij}\cdot \textbf{r}_{ij} < 0 \\ (0, \quad 1) &  otherwise \end{cases}
 
+\\
 \\
 
 p_{ij}^b = \tilde{p}_{ij} \chi_{ij} 
 
+\\
 \\
 
 \tilde{p}_{ij} = max(min(\lambda |p_i + p_j|, \lambda p_{max}), p_{min})
 
+\\
 \\
 
 \chi_{ij}  = \sqrt{\frac{\omega({r}_{ij}, l_0)}{\omega(l_0/2, l_0)}}
 
+\\
 \\
 
 k_{ij} =  \begin{cases} \chi_{ij} & 0.5 \le {r}_{ij}/l_0 < 1 \\ 1 & {r}_{ij}/l_0 < 0.5 \end{cases}

src/gpukernels2d.jl

@@ -531,8 +531,11 @@ Compute ∂ρ∂t - density derivative includind density diffusion.
 
 ```math
 
-\\frac{\\partial \\rho_i}{\\partial t} = \\sum  m_j \\textbf{v}_{ij} \\cdot \\nabla_i W_{ij} + \\delta_{\Phi} h c_0 \\sum \\Psi_{ij} \\cdot \\nabla_i W_{ij} \\frac{m_j}{\\rho_j}
+\\frac{\\partial \\rho_i}{\\partial t} = \\sum  m_j \\textbf{v}_{ij} \\cdot \\nabla_i W_{ij} + \\delta_{\\Phi} h c_0 \\sum \\Psi_{ij} \\cdot \\nabla_i W_{ij} \\frac{m_j}{\\rho_j}
 
+\\\\
+
+\\Psi_{ij} = 2 (\\rho_{ij}^T + \\rho_{ij}^H) \\frac{\\textbf{r}_{ij}}{r_{ij}^2 + \\eta^2}
 ```
 
 """
@@ -901,6 +904,11 @@ end
     ∂v∂t!(∑∂v∂t,  ∇Wₙ, pairs, m, ρ, c₀, γ, ρ₀) 
 
 The momentum equation (without dissipation).
+
+
+\\frac{\\partial \\textbf{v}_i}{\\partial t} = - \\sum  m_j \\left( \\frac{p_i}{\\rho^2_i} + \\frac{p_j}{\\rho^2_j} \\right) \\nabla_i W_{ij}
+
+
 """
 function ∂v∂t!(∑∂v∂t,  ∇W, P, pairs, m, ρ) 
     gpukernel = @cuda launch=false kernel_∂v∂t!(∑∂v∂t,  ∇W, P, pairs, m, ρ)