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+# Fun with compute shaders and fluid dynamics
+
+## First for some theory…
+
+What we want (Navier-Stokes):
+
+$$\begin{aligned} \partial_t  \rho + \nabla \cdot (\rho u) &= 0 \\ \partial_t u + (u \cdot \nabla) u &= -\frac{1}{\rho} \nabla p + 2\nu\nabla \cdot (\mathrm{S})\end{aligned}$$
+
+Pressure $p = c_s^2 \rho$
+
+Kinetic viscosity: $\nu = c_s^2 \tau$
+
+Tensor: $\mathrm{S} = \frac{1}{2} (\nabla u + (\nabla u)^\top)$
+
+What we use (Boltzmann equilibrium):
+
+$$\left( \partial_t + \xi \cdot \partial_x + \frac{F}{\rho} \cdot \partial_\xi \right) f = \Omega(f) \left( = \partial_x f \cdot \frac{dx}{dt} + \partial_\xi f \cdot \frac{d\xi}{dt} + \partial_t f \right)$$
+
+How we get there (BGK LBM):
+
+$$\Omega(f) := -\frac{f-f^\text{eq}}{\tau} \Delta t$$
+
+$$(\partial_t + \xi \cdot \nabla_x) f = -\frac{1}{\tau} (f(x,\xi,t) - f^\text{eq}(x,\xi,t))$$
+
+$$\newcommand{\V}[2]{\begin{pmatrix}#1\\#2\end{pmatrix}} \{\xi_i\}_{i=0}^8 = \left\{ \V{0}{0}, \V{-1}{\phantom{-}1}, \V{-1}{\phantom{-}0}, \V{-1}{-1}, \V{\phantom{-}0}{-1}, \V{\phantom{-}1}{-1}, \V{1}{0}, \V{1}{1}, \V{0}{1} \right\}$$
+
+$$(\partial_t + \xi_i \cdot \nabla_x) f_i(x,t) = -\frac{1}{\tau} (f_i(x,t) - f_i^\text{eq}(x,t))$$
+
+$$f_i^\text{eq} = w_i \rho \left( 1 + \frac{u \cdot \xi_i}{c_s^2} + \frac{(u \cdot \xi_i)^2}{2c_s^4} - \frac{u \cdot u}{2c_s^2} \right)$$
+
+$$\rho(x,t) = \sum_{i=0}^{q-1} f_i(x,t)$$
+
+$$\rho u(x,t) = \sum_{i=0}^{q-1} \xi_i f_i(x,t)$$
+
+$$w_0 = \frac{4}{9}, \ w_{2,4,6,8} = \frac{1}{9}, \ w_{1,3,5,7} = \frac{1}{36}$$
+
+$$\overline{f_i} = f_i + \frac{1}{2\tau}(f_i - f_i^\text{eq})$$
+
+$$\overline\tau = \tau + \frac{1}{2}$$
+
+$$\overline{f_i}(x+\xi_i,t+1) = \overline{f_i}(x,t) - \frac{1}{\overline\tau} (\overline{f_i}(x,t) - f_i^\text{eq}(x,t))$$
+
+$$f_i^\text{out}(x,t) = f_i(x,t) - \frac{1}{\tau}(f_i(x,t) - f_i^\text{eq}(x,t))$$
+
+$$f_i(x+\xi_i,t+1) = f_i^\text{out}(x,t)$$
+
+## …translated into GLSL compute shaders
+
+```cpp
+layout (local_size_x = 1, local_size_y = 1) in;
+
+layout (std430, binding=1) buffer bufferCollide{ float collideCells[]; };
+layout (std430, binding=2) buffer bufferStream{ float streamCells[]; };
+layout (std430, binding=3) buffer bufferFluid{ float fluidCells[]; };
+
+uniform uint nX;
+uniform uint nY;
+```
+
+```cpp
+const uint  q         = 9;
+const float weight[q] = float[](
+	1./36., 1./9., 1./36.,
+	1./9. , 4./9., 1./9. ,
+	1./36 , 1./9., 1./36.
+);
+```
+
+```cpp
+uint indexOfDirection(int i, int j) {
+	return 3*(j+1) + (i+1);
+}
+
+uint indexOfLatticeCell(uint x, uint y) {
+	return q*nX*y + q*x;
+}
+
+/* [...] */
+
+float get(uint x, uint y, int i, int j) {
+	return collideCells[indexOfLatticeCell(x,y) + indexOfDirection(i,j)];
+}
+```
+
+```cpp
+float equilibrium(float d, vec2 v, int i, int j) {
+	return w(i,j) * d * (1 + 3*comp(i,j,v) + 4.5*sq(comp(i,j,v)) - 1.5*sq(norm(v)));
+}
+```
+
+```cpp
+void main() {
+	const uint x = gl_GlobalInvocationID.x;
+	const uint y = gl_GlobalInvocationID.y;
+
+	const float d = density(x,y);
+	const vec2  v = velocity(x,y,d);
+
+	setFluid(x,y,v,d);
+
+	for ( int i = -1; i <= 1; ++i ) {
+		for ( int j = -1; j <= 1; ++j ) {
+			set(x,y,i,j, get(x,y,i,j) + omega * (equilibrium(d,v,i,j) - get(x,y,i,j)));
+		}
+	}
+}
+```
+
+```cpp
+void main() {
+	const uint x = gl_GlobalInvocationID.x;
+	const uint y = gl_GlobalInvocationID.y;
+
+	if ( x != 0 && x != nX-1 && y != 0 && y != nY-1 ) {
+		for ( int i = -1; i <= 1; ++i ) {
+			for ( int j = -1; j <= 1; ++j ) {
+				set(x+i,y+j,i,j, get(x,y,i,j));
+			}
+		}
+	} else {
+		// rudimentary bounce back boundary handling
+		[...]
+	}
+}
+```
+
+## Visuals
+
+![Pleasing snapshot of an artfully amplified implosion](https://static.kummerlaender.eu/media/boltzstern_1.jpg)
+
+![Pleasing snapshot of an artfully amplified implosion](https://static.kummerlaender.eu/media/boltzstern_2.jpg)
+
+![Pleasing snapshot of an artfully amplified implosion](https://static.kummerlaender.eu/media/boltzstern_3.jpg)
+
+## Reaching down from the heavens