From 0042d4e69cf1575dd59d42a2e2579ae597919d41 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Wed, 13 Mar 2024 23:41:28 +0100
Subject: [PATCH] updates
---
.../content/1-dimensionless-lbm.md | 2 +-
sage/hermite.sage | 28 +++++++++++++------
2 files changed, 21 insertions(+), 9 deletions(-)
diff --git a/lattice-boltzmann.com/content/1-dimensionless-lbm.md b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
index 394f79b..b2cb286 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 56b18ab..5a5af1d 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")
--
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