diff --git a/lattice-boltzmann.com/content/1-dimensionless-lbm.md b/lattice-boltzmann.com/content/1-dimensionless-lbm.md index 394f79bd071f1d409fe75f676a9984211081a267..b2cb28646459b3f40d40033e78b15e3ec766e906 100644 --- a/lattice-boltzmann.com/content/1-dimensionless-lbm.md +++ b/lattice-boltzmann.com/content/1-dimensionless-lbm.md @@ -113,7 +113,7 @@ $$ \partial_t^\ast f^\ast+\bm{\xi}^\ast\cdot \bm{\nabla}^\ast f^\ast=-\frac{1}{\mathrm{Kn}}\left(f^\ast-{f^{eq}}^\ast\right), \end{equation} $$ -where the space, time, and microscopic velocity is omitted and where +where the space, time, and microscopic velocity dependence is omitted and where $$ \mathrm{Kn}=\frac{\tau\xi_0}{L}, $$ diff --git a/sage/hermite.sage b/sage/hermite.sage index 56b18ab3ad2bc6555bc2ec9d1fa3d15fdb4a81f7..5a5af1db144f61c07e27a8b0173b4249f5d33cab 100644 --- a/sage/hermite.sage +++ b/sage/hermite.sage @@ -1,16 +1,28 @@ xi = var('xi') u = var('u') theta = var('theta') +xi0 = var('xi0') +n = var('n') assume(theta > 0) -h0 = 1 -h1 = xi -h2 = xi^2 - 1 -h3 = xi^3 - 3*xi -h4 = xi^4 - 6*xi^2 + 3 -h5 = xi^5 - 10*xi^3 + 15*xi +def hn(n): + return (-1)^n * xi0^n * exp(xi^2/(2 * xi0^2))*diff(exp(-xi^2/(2 * xi0^2)), xi, n) -w = 1/sqrt(2*pi) * exp(-xi^2/2) +print("Hermite polynomials") +h0 = hn(0) +print(h0) +h1 = hn(1) +print(h1) +h2 = hn(2).full_simplify() +print(h2) +h3 = hn(3).full_simplify() +print(h3) +h4 = hn(4).full_simplify() +print(h4) +h5 = hn(5).full_simplify() +print(h5) + +w = 1/sqrt(2*pi) * exp(-xi^2/(2*xi0^2)) print("Coefficients") n0 = integrate(w * h0, xi, -infinity, infinity) @@ -27,7 +39,7 @@ n5 = integrate(w * h5 * h5, xi, -infinity, infinity) print(n5.canonicalize_radical()) -bol = 1 / (sqrt(2 * pi * theta)) * exp(-(xi - u)^2/(2 * theta)) +bol = 1 / (sqrt(2 * pi * xi0^2)) * exp(-(xi - u)^2/(2 * xi0^2)) res = integrate(bol, xi, -infinity, infinity) print("Moments of Boltzmann")