update

tex/upwind-flux.tex

@@ -365,6 +365,10 @@ Or maybe just the advection equation.}
 the consistency property when $f$ is a nonlinear function.
 Maybe just use the Burgers equation?}
 
+\todo{\cite[section 3.1.1]{toroRiemannSolversNumerical2013} says that the
+Euler equations in 1+1D with a polytropic equation of state satisfy the
+\textit{homogeneity property}, that is, $f(u) = A(u) u$.}
+
 
 
 \section{Nonlinear case}
@@ -393,7 +397,7 @@ over the computational domain, \eg{} it is given by direct sum of local solution
   q(x) = \bigoplus_{k} q^k(x) \, ,
   \label{eq:global-solution}
 \end{align}
-where each $q^k = q^k(x)$ is smooth.
+where each $q^k = q^k(x)$ is smooth over $x \in [x^k_l, x^k_r]$.
 Let $q^k_{l,r} = q(x^{k}_{l,r})$ and similarly for the other domains.
 Note that in general $q^k_r \neq q^{k+1}_r$ and so on.
 The averaged state and its deviation can now be defined in the $k$th subdomain by
@@ -468,7 +472,7 @@ The corresponding weak formulation of these equations then read
   \braket{\pdvt \delta q}{\psi}_D - \braket{f'(q_0)\delta q}{\pdvx \psi}_D + \eval{\qty[n^- (f'(q_0) \delta q)^\ast \psi]}_{\partial D} = 0 \, .
   \label{eq:weak_form_deviation}
 \end{align}
-On first sight this appears to be an unnecessary compliation, because one now has to
+On first sight this appears to be an unnecessary complication, because one now has to
 solve two conservation laws and, thus, design two numerical flux functions,
 $n^- f^\ast$ and $n^- (f(q_0)\delta q)^\ast$, in order to establish the next-neighbor coupling
 and stability.
@@ -484,7 +488,7 @@ we immediately know what the value on the opposing side of the boundary has to b
 of $q_0$. Hence, the next-neighbor coupling is automatically resolved and one does not have
 to think about interior and exterior states, \ie{}, $q^-_0 = q^+_0$.
 The consistency property for numerical fluxes then immediately suggests the choice $n^- f^\ast = n^- f$,
-which then is also satisfies the compatibility property.
+which then also satisfies the compatibility property.
 
 Because \eqref{eq:weak_form_average} is independent of $q_0$,
 we can in principle evolve $q_0$ before we evolve $\delta q$, and which we now assume
@@ -518,12 +522,12 @@ neglecting the higher order corrections on the RHS,
   + \braket{\pdvt \delta q}{\psi}_D - \braket{f'(q_0)\delta q}{\pdvx \psi}_D + \eval{\qty[n^- (f'(q_0) \delta q)^\ast \psi]}_{\partial D}
   \, .
 \end{align}
-Note that this does formulation agrees with the linear combination of
+Note that this formulation agrees with the linear combination of
 \eqref{eq:weak_form_average} and \eqref{eq:weak_form_deviation}.
 Furthermore, this formulation agrees with the weak formulation
 \eqref{eq:weak_form_average}
-while also assuming the frozen coefficient approximation for $F(q_0, \delta q)$ from before,
-This agrees now \textit{almost} agrees with a weak formulation
+while also assuming the frozen coefficient approximation for $F(q_0, \delta q)$ from before.
+This now \textit{almost} agrees with a weak formulation
 of \eqref{eq:weak_form} up to $\mathcal{O}(\delta q^2)$.
 
 We now assert that the following numerical flux function, which is a combination
@@ -537,9 +541,9 @@ upwinding numerical flux function \eqref{eq:num_flux_linear},
   &\quad + \frac{1}{2} \left( 1 + \frac{n^- f}{| n^- f |} \right) ( n^- f(q_0) + n^- F^{(+)}(q_0,\delta q^-) - n^- F^{(-)}(q_0,\delta q^+)) \, ,
   \label{eq:num_flux_nonlinear}
 \end{align}
-where $n- F^{(\pm)}(q_0,\delta q) = A^{n^-(\pm)} \delta q = n^- f^{(\pm)}(q_0) \delta q$ and so on.
+where $n^- F^{(\pm)}(q_0,\delta q) = A^{n^-(\pm)} \delta q = n^- f^{(\pm)}(q_0) \delta q$ and so on.
 A few comments are in order.
-First, the anatomy of \eqref{eq:num_flux_nonlinear} is very similar to \eqref{eq:num_flux_nonlinear},
+First, the anatomy of \eqref{eq:num_flux_nonlinear} is very similar to \eqref{eq:num_flux_linear},
 \eg{} we have two prefactors $\frac{1}{2} \left( 1 \mp \frac{n^- f}{| n^- f |} \right)$
 that control when to use the right upwinding terms.
 Second, The difference between those two formulae lies in the upwinding fluxes.