Update docs

demo/OrrSommerfeld_eigs.py

@@ -76,6 +76,10 @@ def assemble(self, scale=None):
         v = TestFunction(test)
         u = TrialFunction(trial)
 
+        Re = self.Re
+        a = self.alfa
+        g = 1j*a*Re
+
         # (u'', v)_w
         K = inner(v, Dx(u, 0, 2))
 
@@ -91,11 +95,18 @@ def assemble(self, scale=None):
         # (u, v)_w
         M = inner(v, u)
 
-        Re = self.Re
-        a = self.alfa
         B = -Re*a*1j*(K-a**2*M)
         A = Q-2*a**2*K+(a**4 - 2*a*Re*1j)*M - 1j*a*Re*(K2-a**2*K1)
+
+        # Alternatively:
+        #A1 = lambda f: Dx(f, 0, 4)-2*a**2*Dx(f, 0, 2)+(a**4-2*g)*f-g*(1-x**2)*(Dx(f, 0, 2)-f)
+        #B1 = lambda f: -g*(Dx(f, 0, 2)-f)
+        #A = inner(A1(u), v)
+        #B = inner(B1(u), v)
+        #A = sum(A[1:], A[0])
+        #B = sum(B[1:], B[0])
         A, B = A.diags().toarray(), B.diags().toarray()
+
         if scale is not None and not (scale[0] == 0 and scale[1] == 0):
             assert isinstance(scale, tuple)
             assert len(scale) == 2
@@ -147,7 +158,7 @@ def get_eigval(nx, eigvals, verbose=False):
 if __name__ == '__main__':
     import argparse
     parser = argparse.ArgumentParser(description='Orr Sommerfeld parameters')
-    parser.add_argument('--N', type=int, default=120,
+    parser.add_argument('--N', type=int, default=400,
                         help='Number of discretization points')
     parser.add_argument('--Re', default=8000.0, type=float,
                         help='Reynolds number')

docs/demos/DrivenCavity/drivencavity.do.txt

@@ -41,7 +41,7 @@ The nonlinear steady Navier Stokes equations are given in strong form as
 !et
 where $\bs{u}, p$ and $\nu$ are, respectively, the
 fluid velocity vector, pressure and kinematic viscosity. The domain
-$\Omega = [-1, 1]^2$ and the nonlinear term $\bs{u} \bs{u}$ is the
+$\Omega = (-1, 1)^2$ and the nonlinear term $\bs{u} \bs{u}$ is the
 outer product of vector $\bs{u}$ with itself. Note that the final
 $\int_{\Omega} p dx = 0$ is there because there is no Dirichlet boundary
 condition on the pressure and the system of equations would otherwise be

docs/demos/KleinGordon/kleingordon.do.txt

@@ -65,7 +65,7 @@ t=0)}{\partial t} = u_t^0. label{eq:init}
 The spatial coordinates are here denoted as $\boldsymbol{x} = (x, y, z)$, and
 $t$ is time. The parameter $\gamma=\pm 1$ determines whether the equations are focusing
 ($+1$) or defocusing ($-1$) (in the movie we have used $\gamma=1$).
-The domain $\Omega=[-2\pi, 2\pi]^3$ is triply
+The domain $\Omega=[-2\pi, 2\pi)^3$ is triply
 periodic and initial conditions will here be set as
 
 !bt

docs/demos/KuramatoSivashinsky/kuramatosivashinsky.do.txt

@@ -34,12 +34,12 @@ FIGURE: [https://rawgit.com/spectralDNS/spectralutilities/master/movies/Kuramato
 
 The Kuramato-Sivashinsky (KS) equation is known for its chaotic bahaviour, and it is
 often used in study of turbulence or turbulent combustion. We will here solve
-the KS equation in a doubly periodic domain $[-30\pi, 30\pi]^2$, starting from a
+the KS equation in a doubly periodic domain $\Omega=[-30\pi, 30\pi)^2$, starting from a
 single Gaussian pulse
 !bt
 \begin{align}
 \frac{\partial u(\bs{x},t)}{\partial t} &+ \nabla^2 u(\bs{x},t) + \nabla^4
-u(\bs{x},t) + |\nabla u(\bs{x},t)|^2 = 0 \quad \text{for }\, \bs{x} \in \Omega=[-30 \pi, 30\pi]^2
+u(\bs{x},t) + |\nabla u(\bs{x},t)|^2 = 0 \quad \text{for }\, \bs{x} \in \Omega
 label{eq:ks} \\
 u(\bs{x}, 0) &= \exp(-0.01 \bs{x} \cdot \bs{x}) \notag
 \end{align}

docs/demos/Moebius/moebius.do.txt

@@ -53,9 +53,9 @@ regular periodic boundary conditions
 \end{equation}
 !et
 will apply cite{Kalvoda2020}, and we can discretize using Fourier
-exponentials in the $\theta$-direction. Since the reference domain of Fourier
-exponentials is $[-\pi, \pi]$ we define $\varphi = \theta /(2 R)$, such that
-the computational domain is $(\varphi, t) \in \mathbb{I}^2 = [-\pi, \pi] \times [-1, 1]$,
+exponentials in the $\theta$-direction. Since the reference domain of periodic Fourier
+exponentials is $[-\pi, \pi)$ we define $\varphi = \theta /(2 R)$, such that
+the computational domain is $(\varphi, t) \in \mathbb{I}^2 = [-\pi, \pi) \times (-1, 1)$,
 with the parametrization
 
 !bt
@@ -65,7 +65,7 @@ with the parametrization
     z(\varphi, t) &= -t \sin {\varphi}.
 \end{align}
 !et
-Note that the reference domain corresponds to $(\theta, t) \in = [-2\pi R, 2\pi R] \times [-1, 1]$.
+Note that the reference domain corresponds to $(\theta, t) \in = [-2\pi R, 2\pi R) \times (-1, 1)$.
 It is also trivial to adjust the $t$-domain with an affine map for a more narrow or
 wider strip, but this added complexity is not discussed here. One can simply
 choose the width of the strip below when choosing function space for the

docs/demos/Poisson/poisson.do.txt

@@ -31,7 +31,7 @@ Poisson's equation is given as
 
 !bt
 \begin{align}
-\nabla^2 u(x) &= f(x) \quad \text{for }\, x \in [-1, 1], label{eq:poisson}\\
+\nabla^2 u(x) &= f(x) \quad \text{for }\, x \in (-1, 1), label{eq:poisson}\\
 u(-1)&=a, u(1)=b, \notag
 \end{align}
 !et

docs/demos/Poisson3D/poisson3d.do.txt

@@ -33,7 +33,7 @@ u(x, y, 2\pi) &= u(x, y, 0),
 \end{align}
 !et
 where $u(\bs{x})$ is the solution and $f(\bs{x})$ is a function. The domain
-$\Omega = [-1, 1]\times [0, 2\pi]^2$.
+$\Omega = (-1, 1)\times [0, 2\pi)^2$.
 
 To solve Eq. (ref{eq:3d:poisson}) with the Galerkin method we will make use of
 smooth basis functions, $v(\bs{x})$, that satisfy the given boundary

docs/demos/PolarHelmholtz/polarhelmholtz.do.txt

@@ -39,7 +39,7 @@ We use polar coordinates $(\theta, r)$, defined as
 \end{align}
 !et
 
-which leads to a Cartesian product mesh $(\theta, r) \in [0, 2\pi) \times [0, a]$
+which leads to a Cartesian product mesh $(\theta, r) \in [0, 2\pi) \times (0, a)$
 suitable for numerical implementations. Note that the
 two directions are ordered with $\theta$ first and then $r$, which is less common
 than $(r, \theta)$. This has to do with the fact that we will need to

docs/demos/RayleighBenard/rayleighbenard.do.txt

@@ -30,7 +30,7 @@ TOC: off
 ===== The Rayleigh Bénard equations =====
 label{demo:rayleighbenard}
 
-The governing equations solved in domain $\Omega=[-1, 1]\times [0, 2\pi]$ are
+The governing equations solved in domain $\Omega=(-1, 1)\times [0, 2\pi)$ are
 
 !bt
 \begin{align}

docs/demos/Stokes/stokes.do.txt

@@ -33,7 +33,7 @@ Stokes' equations are given in strong form as
 !et
 where $\bs{u}$ and $p$ are, respectively, the
 fluid velocity vector and pressure, and the domain
-$\Omega = [0, 2\pi]^2 \times [-1, 1]$. The flow is assumed periodic
+$\Omega = [0, 2\pi)^2 \times (-1, 1)$. The flow is assumed periodic
 in $x$ and $y$-directions, whereas there is a no-slip homogeneous Dirichlet
 boundary condition on $\bs{u}$ on the boundaries of the $z$-direction, i.e.,
 $\bs{u}(x, y, \pm 1) = (0, 0, 0)$. (Note that we can configure shenfun with

docs/demos/Tau/tau.do.txt

@@ -33,11 +33,11 @@ TOC: off
 
 ===== The tau method for Poisson's equation in 1D =====
 
-Poisson's equation is given on a domain $[-1, 1]$ as
+Poisson's equation is given on a domain $\Omega = (-1, 1)$ as
 
 !bt
 \begin{align}
-\nabla^2 u(x) &= f(x) \quad \text{for }\, x \in [-1, 1], label{eq:poisson}\\
+\nabla^2 u(x) &= f(x) \quad \text{for }\, x \in \Omega, label{eq:poisson}\\
 u(-1)&=a, u(1)=b, \notag
 \end{align}
 !et

docs/source/drivencavity.rst

@@ -49,7 +49,7 @@ The nonlinear steady Navier Stokes equations are given in strong form as
 
 where :math:`\boldsymbol{u}, p` and :math:`\nu` are, respectively, the
 fluid velocity vector, pressure and kinematic viscosity. The domain
-:math:`\Omega = [-1, 1]^2` and the nonlinear term :math:`\boldsymbol{u} \boldsymbol{u}` is the
+:math:`\Omega = (-1, 1)^2` and the nonlinear term :math:`\boldsymbol{u} \boldsymbol{u}` is the
 outer product of vector :math:`\boldsymbol{u}` with itself. Note that the final
 :math:`\int_{\Omega} p dx = 0` is there because there is no Dirichlet boundary
 condition on the pressure and the system of equations would otherwise be