35.9.1. Scalar non-linear equation in 1-dimensional domain
The hyperbolic equation for \(u(x,t)\), \(\partial_t u + \partial_x F(u) = 0\), is often the inviscid limit of the 2-nd order equation
\[\partial_t u + \partial_x F(u) - \partial_x \left( \varepsilon \partial_x u \right) = 0 \ .\]
Term \(\varepsilon \partial_x u\) represents a diffusive flux, and the inviscid limit is defined by the condition \(\varepsilon \rightarrow 0\).
Definition of an entropy function \(\eta(u)\), and its flux \(q(u)\). Let \(\eta(u)\) be a function of the field \(u\). Evaluating its time derivative using the viscous equation,
\[\begin{split}\begin{aligned}
\partial_t \eta(u)
& = \eta'(u) \partial_t u = \\
& = \eta'(u) \left[ - \partial_x F(u) - \partial_x ( \varepsilon \partial_x u ) \right] = \\
& = - \eta'(u) F'(u) \partial_x u - \eta'(u) ( \varepsilon u_{/x} )_{/x} \ .
\end{aligned}\end{split}\]
If a function \(q(u)\) satisfies the condition
\[q'(u) = \eta'(u) F'(u) \ ,\]
function \(\eta(u)\) can be defined as an entropy function and \(q(u)\) as the corresponding entropy flux.
Differential inequality. Let these functions exist. Then
\[\partial_t \eta + \partial_x q = - \eta' \left( \varepsilon u_{/x} \right)_{/x} \ ,\]
is a differential balance equation for the entropy. The last term can be manipulated to find
\[\begin{split}\begin{aligned}
0
& = \partial_t \eta + \partial_x q - \eta' \left( \varepsilon u_{/x} \right)_{/x} = \\
& = \partial_t \eta + \partial_x q - \left( \varepsilon \eta'(u) u_{/x} \right)_{/x} + \eta''(u) \varepsilon \left( u_{/x} \right)^2 \ ,
\end{aligned}\end{split}\]
For \(\varepsilon \ge 0\), for convex entropy function \(\eta''(u)\), a differential inequality follows
\[
\partial_t \eta + \partial_x q + \left( \varepsilon \eta'(u) u_{/x} \right)_{/x} \le 0 \ .
\]
Integral inequality and jump conditions. Differential inequality is the local counterpart of the integral equation for a volume \(V\) at rest
\[\frac{d}{dt} \int_{V} \eta + \oint_{\partial V} n_x \, q + \oint_{\partial V} \varepsilon q'(u) u_{/x} n_x \le 0 \ ,\]
or for a generic volume \(v_t\),
\[\frac{d}{dt} \int_{v_t} \eta - \oint_{\partial v_t} n_x v_b \eta + \oint_{\partial v_t} n_x \, q + \oint_{\partial v_t} \varepsilon q'(u) u_{/x} n_x \le 0 \ ,\]
being \(v_b\) the velocity of the points of the boundary of the domain \(\partial v_t\). Jump condition in the inviscid limit \(\varepsilon \rightarrow 0\) follows from the collapse of the volume on a surface (\(\frac{|v_t|}{|\partial v_t|} \rightarrow 0\)), and from the arbitrariness of this surface
\[v_b [ \eta ] \ge [ q ] \ .\]
Weak shocks. Expanding in Taylor series jump condition for weak shocks provides an alternative version of the entropy condition, as a relation between the eigenvalues of the system and the shock speed. The shock speed reads \(\dot{s} = \frac{[f]}{[u]} = \frac{f_R-f_L}{u_R - u_L}\). Taylor expansion of jump condition relies on Taylor expansion of the jumps there in and of the speed of the shock
\[\begin{split}\begin{aligned}
f_R - f_L & = f'(L) \Delta u + f''(L) \frac{\Delta u^2}{2} + f'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\
\dot{s} & = f'(L) + f''(L) \frac{\Delta u}{2} + f'''(L) \frac{\Delta u^2}{6} + o(\Delta u^2) \\
\eta_R - \eta_L & = \eta'(L) \Delta u + \eta''(L) \frac{\Delta u^2}{2} + \eta'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\
q_R - q_L & = q'(L) \Delta u + q''(L) \frac{\Delta u^2}{2} + q'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\
\end{aligned}\end{split}\]
and comparing terms of the same order
\[\begin{aligned}
0 \le \Delta u ( \eta'_L f'_L - q'_L ) + \Delta u^2 \left( \frac{1}{2} f'_L \eta''_L + \frac{1}{2} f''_L \eta'_L - \frac{1}{2} q''_L \right) + \Delta u^3 \left( \frac{1}{6} \eta'''_L f'_L + \frac{1}{4} \eta''_L f''_L + \frac{1}{6} \eta'_L f'''_L - \frac{1}{6} q'''_L \right) + o(\Delta u^3) \ .
\end{aligned}\]
Exploiting the relation \(q' = \eta' f'\), and evaluating higher-order derivatives
\[\begin{split}\begin{aligned}
q'' & = \eta'' f' + \eta' f'' \\
q'''& = \eta''' f' + 2 \eta'' f'' + \eta' f''' \\
\end{aligned}\end{split}\]
it follows that the first two terms of the jump condition inequality are identically zero, so that it can be written as
\[0 \le -\frac{1}{12} \Delta u^3 \eta''_L f''_L \ .\]
For a system with convex entropy function \(\eta''(u) \ge 0\), and convex flux \(f''(u) \ge 0\), it follows that \(\Delta u \le 0\), i.e. \(u_R < u_L\). Using the Taylor expansion (either centered in the left or right state) of the speed of the shock,
\[\begin{split}\begin{aligned}
\dot{s} & = f'_L + f''_L \frac{\Delta u}{2} \le f'_L \\
\dot{s} & = f'_R - f''_R \frac{\Delta u}{2} \ge f'_R \\
\end{aligned}\end{split}\]
the relation between the eigenvalues of the systems in the left \(\lambda_L = f'(u_L)\) and right state \(\lambda_R = f'(u_R)\), and the speed of the shock immediately follows
\[\lambda_L \ge \dot{s} \ge \lambda_R \ .\]
35.9.1.1. Burgers’ equation
35.9.2. Non-linear system of equations in 1-dimensional domain
\[\partial_t \mathbf{u} + \partial_x \mathbf{F}(\mathbf{u}) = \mathbf{0} \ .\]
If a shock originates from the \(i^{th}\) characteristic family, the condition \(\lambda_{i,L} \ge \dot{s}_i \ge \lambda_{i,R}\) holds.
Let \(\mathbf{A}(\mathbf{u}) = \nabla_{\mathbf{u}} \mathbf{F}(\mathbf{u})\) the convection matrix, \(\mathbf{A} = \mathbf{R} \boldsymbol\Lambda \mathbf{L}\) its spectral decomposition, and \(\mathbf{v}\) the characteristic variables s.t. \(d \mathbf{v} = \mathbf{L} d \mathbf{u}\), or \(d \mathbf{u} = \mathbf{R} d \mathbf{v}\).
The convective form of the equation reads
\[\partial_t \mathbf{u} + \mathbf{A} \partial_x \mathbf{u} = \mathbf{0} \ ,\]
or in diagonal form using characteristic variables,
\[\partial_t \mathbf{v} + \boldsymbol\Lambda \partial_x \mathbf{v} = \mathbf{0} \ .\]
A shock originating from the \(i^{th}\) family of characteristics comes from the scalar equation
\[\partial_t v_i + \lambda_i \partial_x v_i = 0 \ .\]
The speed of the shock satisfies \(\dot{s}_i [ \mathbf{u} ]_i = [ \mathbf{F}(\mathbf{u}) ]_i\), with \([ \mathbf{u} ]_i = \mathbf{u}_i^+ - \mathbf{u}_i^-\), and
\[\begin{split}\begin{aligned}
\mathbf{u}_i^- & = \mathbf{u}(\mathbf{v}_{\notin i}, v_i^-) \\
\mathbf{u}_i^+ & = \mathbf{u}(\mathbf{v}_{\notin i}, v_i^+) \\
\end{aligned}\end{split}\]
Taylor expansion of the relation gives (no sum on \(i\), as \(\Delta v_i\) is the only non-zero component of \(\Delta \mathbf{v}\) across a shock originating from the \(i^{th}\) family of characteristics)
\[\begin{split}\begin{aligned}
\dot{s}_i \Delta_i u_k
& = \Delta_i F_k = \\
& = \partial_{u_\ell} F_k \Delta_i u_\ell + \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_{\ell} \Delta_i u_{m} = \\
& = A_{k \ell} R_{\ell i} \Delta v_i + \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_{\ell} \Delta_i u_{m}
\end{aligned}\end{split}\]
Exploiting spectral decomposition of the convection matrix
\[\left( \dot{s}_i R_{ki} - R_{km} \Lambda_{mn} \underbrace{L_{n\ell} R_{\ell i}}_{\delta_{ni}} \right) \Delta v_i = \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_\ell \Delta_i u_m\]
\[\left( \dot{s}_i - \lambda_i \right) R_{ki}\Delta v_i = \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_\ell \Delta_i u_m\]
\[\left( \dot{s}_i - \lambda_i \right) \Delta_i u_k = \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_\ell \Delta_i u_m \ .\]
If the flux is convex (here the flux is a vector quantity. Is this the requirement for each of its components \(F_k\)?), i.e. the Hessian is (semi)definite positive, the right-hand side is always non-negative.
The following relations have been used above
\[d u_l = R_{lm} d v_m \ .\]
\[\Delta_i u_l = \partial_{v_m} u_l \underbrace{\Delta_i v_m}_{= \delta_{im} \Delta v_i} = \partial_{v_i} u_l \Delta v_i = R_{li} \Delta v_i \ .\]
Let \(\eta(\mathbf{u})\) the entropy function of the system \(\partial_t \mathbf{u} + \partial_x \mathbf{F}(\mathbf{u}) = \mathbf{0}\). It follows
\[\begin{split}\begin{aligned}
\partial_t \eta
& = \partial_{u_i} \eta \, \partial_t u_i \\
& = - \partial_{u_i} \eta \, \partial_{x} F_i = \\
& = - \partial_{u_i} \eta \, \partial_{u_j} F_i \, \partial_x u_j = \\
& = - \partial_{u_j} q \, \partial_x u_j = \\
& = - \partial_x q \ ,
\end{aligned}\end{split}\]
i.e. the compatibility equation between the entropy function \(\eta(\mathbf{u})\) and the entropy flux \(q(\mathbf{u})\) reads
\[\partial_{u_j} q = \partial_{u_j} F_i \, \partial_{u_i} \eta = A_{ij} \partial_{u_i} \eta \ ,\]
or using vector formalism
\[\nabla_{\mathbf{u}} q = \nabla_{\mathbf{u}} \mathbf{F} \cdot \nabla_{\mathbf{u}} \eta = \mathbf{A} \cdot \nabla_{\mathbf{u}} \eta \ .\]
Let \(\partial_t \mathbf{u} + \partial_x \mathbf{F}(\mathbf{u}) = \partial_x( \mathbf{D} \partial_x \mathbf{u} )\) the diffusive equation. The entropy equation becomes
\[\begin{split}\begin{aligned}
\partial_t \eta
& = \partial_{u_i} \eta \, \partial_t u_i \\
& = - \partial_{u_i} \eta \, \partial_{x} F_i + \partial_{u_i} \eta \partial_x \left( D_{ik} \partial_x u_k \right) = \\
& = - \partial_{u_i} \eta \, \partial_{u_j} F_i \, \partial_x u_j + \partial_x \left( \partial_{u_i} \eta D_{ik} \partial_x u_k \right) - \partial_x \partial_{u_i} \eta D_{ik} \partial_{x} u_k = \\
& = - \partial_{u_j} q \, \partial_x u_j + \partial_x \left( \partial_{u_i} \eta D_{ik} \partial_x u_k \right) - \partial_x u_l \partial_{u_l u_i} \eta D_{ik} \partial_{x} u_k \ .
\end{aligned}\end{split}\]
By definition, if the diffusion term is compatible with the entropy function, the matrix (is this a matrix, a tensor or a matrix of tensors?) \(\nabla_{\mathbf{u} \mathbf{u}} \cdot \mathbf{D}\) is semi-definite positive. If the diffusion term is compatible with the entropy function, in the inviscid limit \(D_{ik} \rightarrow 0\), the entropy inequality becomes
\[\partial_t \eta - \partial_x q \le 0 \ .\]
Now, let the entropy variables be \(\mathbf{w} := \nabla_{\mathbf{u}} \eta\), \(w_i = \partial_{u_i} \eta\). The last term becomes
\[\begin{split}\begin{aligned}
\frac{\partial u_l}{\partial x} \frac{\partial u_k}{\partial x} \frac{\partial}{\partial u_l} \frac{\partial}{\partial u_i} \eta D_{ik}
& = \frac{\partial u_l}{\partial x} \frac{\partial u_k}{\partial x} \frac{\partial}{\partial u_l} w_i D_{ik} = \\
& = \frac{\partial w_i}{\partial x} D_{ik} \frac{\partial u_k}{\partial x} = \\
& = \partial_x \mathbf{w} \cdot \mathbf{D} \cdot \left( \nabla_{\mathbf{u} \mathbf{u}} \eta \right)^{-1} \cdot \partial_x \mathbf{w} \ ,
\end{aligned}\end{split}\]
being
\[\partial_x \mathbf{u} = \partial_x \mathbf{w} \cdot \nabla_{\mathbf{w}} \mathbf{u} = \partial_x \mathbf{w} \cdot \left( \nabla_{\mathbf{u}} \mathbf{w} \right)^{-1} = \partial_x \mathbf{w} \cdot \left( \nabla_{\mathbf{u} \mathbf{u}} \eta \right)^{-1}\]
or, by symmetry of the Hessian, \(\partial_x \mathbf{u} = \left( \nabla_{\mathbf{u} \mathbf{u}} \eta \right)^{-1} \cdot \partial_x \mathbf{w}\).
Compatibility between the diffusion and the entropy function needs \(\mathbf{D} \cdot \left( \nabla_{\mathbf{u} \mathbf{u}} \eta \right)^{-1}\) to be semi-definite positive.
\[\partial_x u_k = \partial_{w_j} u_k \partial_x w_j = \left[ \partial_{u_k} w_j \right]^{-1} \partial_x w_j \ ,\]
\[\delta_{ij} = \partial_{u_i} w_k \partial_{w_k} u_j \ .\]
35.9.2.1. P-system
A common choice of entropy function in a P-system is the mechanical energy (the “reversible” part of the energy).
\[\eta = \rho \left( \frac{1}{2} u^2 + a^2 \ln \frac{\rho}{\rho_0} \right) \ .\]
This quantity decreases across shocks. Corresponding flux reads
\[q = \rho \left( \frac{1}{2} u^2 + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) u \ .\]
Taking the time derivative of the entropy function,
\[\begin{split}\begin{aligned}
\eta_{/t}
& = \rho_{/t} \frac{u^2}{2} + \rho u u_{/t} + a^2 \rho_{/t} \ln \frac{\rho}{\rho_0} + a^2 \rho_{/t} = \\
& = \left( \frac{u^2}{2} + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) \rho_{/t} + u \rho u_{/t} = \\
& = - \left( \frac{u^2}{2} + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) (\rho u)_{/x} - u \left( \rho u u_{/x} + a^2 \rho_{/x} \right) = \\
& = - \left( \rho \frac{u^2}{2} u \right)_{/x} \underbrace{- a^2 \ln \frac{\rho}{\rho_0} \rho u_{/x} - a^2 \ln \frac{\rho}{\rho_0} u \rho_{/x} - a^2 \rho_{/x} u}_{= - \left( a^2 \rho \ln \frac{\rho}{\rho_0} u \right)_{/x}} \underbrace{- a^2 \rho u_{/x} - a^2 \rho_{/x} u}_{= - ( a^2 \rho u )_{/x}} = \\
& = - \left[ \left( \rho \frac{u^2}{2} + a^2 \rho \ln \frac{\rho}{\rho_0} + \rho a^2 \right) u \right]_{/x} \ .
\end{aligned}\end{split}\]
35.9.2.2. Shallow water
35.9.2.3. Euler equations
