(pde:hyperbolic:entropy)= # Physical solution in hyperbolic problems ```{dropdown} Topics :open: This section deals with: - existence of infinite number of solution of hyperbolic problems, and existence of one physical solution of the problem as the inviscid limit of the solution of the diffusive problem - definition of an entropy function, $\eta(u)$, and its flux, $q(\eta)$ - properties of the entropy functions, and fluxes: **convex**, and non-convex fluxes - discussion of the general case, and examples (Euler equations, P-sys, shallow water) ``` (pde:hyperbolic:entropy:1d-scalar)= ## 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, $$\begin{aligned} \partial_t \eta(u) & = \eta'(u) \partial_t u = \\ & = \eta'(u) \left[ - \partial_x F(u) - \partial_x ( \varepsilon \partial_x u ) \right] = \\ & = - \eta'(u) F'(u) \partial_x u - \eta'(u) ( \varepsilon u_{/x} )_{/x} \ . \end{aligned}$$ 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 $$\begin{aligned} 0 & = \partial_t \eta + \partial_x q - \eta' \left( \varepsilon u_{/x} \right)_{/x} = \\ & = \partial_t \eta + \partial_x q - \left( \varepsilon \eta'(u) u_{/x} \right)_{/x} + \eta''(u) \varepsilon \left( u_{/x} \right)^2 \ , \end{aligned}$$ 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 $$\begin{aligned} f_R - f_L & = f'(L) \Delta u + f''(L) \frac{\Delta u^2}{2} + f'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\ \dot{s} & = f'(L) + f''(L) \frac{\Delta u}{2} + f'''(L) \frac{\Delta u^2}{6} + o(\Delta u^2) \\ \eta_R - \eta_L & = \eta'(L) \Delta u + \eta''(L) \frac{\Delta u^2}{2} + \eta'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\ q_R - q_L & = q'(L) \Delta u + q''(L) \frac{\Delta u^2}{2} + q'''(L) \frac{\Delta u^3}{6} + o(\Delta u^3) \\ \end{aligned}$$ 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 $$\begin{aligned} q'' & = \eta'' f' + \eta' f'' \\ q'''& = \eta''' f' + 2 \eta'' f'' + \eta' f''' \\ \end{aligned}$$ 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, $$\begin{aligned} \dot{s} & = f'_L + f''_L \frac{\Delta u}{2} \le f'_L \\ \dot{s} & = f'_R - f''_R \frac{\Delta u}{2} \ge f'_R \\ \end{aligned}$$ 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 $$\lambda_L \ge \dot{s} \ge \lambda_R \ .$$ (pde:hyperbolic:entropy:1d-scalar:burgers)= ### Burgers' equation (pde:hyperbolic:entropy:1d-system)= ## 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. ```{dropdown} Details :open: 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 $$\begin{aligned} \mathbf{u}_i^- & = \mathbf{u}(\mathbf{v}_{\notin i}, v_i^-) \\ \mathbf{u}_i^+ & = \mathbf{u}(\mathbf{v}_{\notin i}, v_i^+) \\ \end{aligned}$$ 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) $$\begin{aligned} \dot{s}_i \Delta_i u_k & = \Delta_i F_k = \\ & = \partial_{u_\ell} F_k \Delta_i u_\ell + \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_{\ell} \Delta_i u_{m} = \\ & = A_{k \ell} R_{\ell i} \Delta v_i + \frac{1}{2} \partial_{u_\ell u_m} F_k \Delta_i u_{\ell} \Delta_i u_{m} \end{aligned}$$ 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 $$d u_l = R_{lm} d v_m \ .$$ $$\Delta_i u_l = \partial_{v_m} u_l \underbrace{\Delta_i v_m}_{= \delta_{im} \Delta v_i} = \partial_{v_i} u_l \Delta v_i = R_{li} \Delta v_i \ .$$ ``` ```{dropdown} Entropy function for systems :open: Let $\eta(\mathbf{u})$ the entropy function of the system $\partial_t \mathbf{u} + \partial_x \mathbf{F}(\mathbf{u}) = \mathbf{0}$. It follows $$\begin{aligned} \partial_t \eta & = \partial_{u_i} \eta \, \partial_t u_i \\ & = - \partial_{u_i} \eta \, \partial_{x} F_i = \\ & = - \partial_{u_i} \eta \, \partial_{u_j} F_i \, \partial_x u_j = \\ & = - \partial_{u_j} q \, \partial_x u_j = \\ & = - \partial_x q \ , \end{aligned}$$ 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 $$\nabla_{\mathbf{u}} q = \nabla_{\mathbf{u}} \mathbf{F} \cdot \nabla_{\mathbf{u}} \eta = \mathbf{A} \cdot \nabla_{\mathbf{u}} \eta \ .$$ ``` ```{dropdown} Entropy function in diffusive equations :open: 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 $$\begin{aligned} \partial_t \eta & = \partial_{u_i} \eta \, \partial_t u_i \\ & = - \partial_{u_i} \eta \, \partial_{x} F_i + \partial_{u_i} \eta \partial_x \left( D_{ik} \partial_x u_k \right) = \\ & = - \partial_{u_i} \eta \, \partial_{u_j} F_i \, \partial_x u_j + \partial_x \left( \partial_{u_i} \eta D_{ik} \partial_x u_k \right) - \partial_x \partial_{u_i} \eta D_{ik} \partial_{x} u_k = \\ & = - \partial_{u_j} q \, \partial_x u_j + \partial_x \left( \partial_{u_i} \eta D_{ik} \partial_x u_k \right) - \partial_x u_l \partial_{u_l u_i} \eta D_{ik} \partial_{x} u_k \ . \end{aligned}$$ 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 $$\partial_t \eta - \partial_x q \le 0 \ .$$ ``` ```{dropdown} Entropy variables Now, let the **entropy variables** be $\mathbf{w} := \nabla_{\mathbf{u}} \eta$, $w_i = \partial_{u_i} \eta$. The last term becomes $$\begin{aligned} \frac{\partial u_l}{\partial x} \frac{\partial u_k}{\partial x} \frac{\partial}{\partial u_l} \frac{\partial}{\partial u_i} \eta D_{ik} & = \frac{\partial u_l}{\partial x} \frac{\partial u_k}{\partial x} \frac{\partial}{\partial u_l} w_i D_{ik} = \\ & = \frac{\partial w_i}{\partial x} D_{ik} \frac{\partial u_k}{\partial x} = \\ & = \partial_x \mathbf{w} \cdot \mathbf{D} \cdot \left( \nabla_{\mathbf{u} \mathbf{u}} \eta \right)^{-1} \cdot \partial_x \mathbf{w} \ , \end{aligned}$$ 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. $$\partial_x u_k = \partial_{w_j} u_k \partial_x w_j = \left[ \partial_{u_k} w_j \right]^{-1} \partial_x w_j \ ,$$ $$\delta_{ij} = \partial_{u_i} w_k \partial_{w_k} u_j \ .$$ ``` (pde:hyperbolic:entropy:1d-system:p-sys)= ### 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 $$q = \rho \left( \frac{1}{2} u^2 + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) u \ .$$ ```{dropdown} Details :open: Taking the time derivative of the entropy function, $$\begin{aligned} \eta_{/t} & = \rho_{/t} \frac{u^2}{2} + \rho u u_{/t} + a^2 \rho_{/t} \ln \frac{\rho}{\rho_0} + a^2 \rho_{/t} = \\ & = \left( \frac{u^2}{2} + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) \rho_{/t} + u \rho u_{/t} = \\ & = - \left( \frac{u^2}{2} + a^2 \ln \frac{\rho}{\rho_0} + a^2 \right) (\rho u)_{/x} - u \left( \rho u u_{/x} + a^2 \rho_{/x} \right) = \\ & = - \left( \rho \frac{u^2}{2} u \right)_{/x} \underbrace{- a^2 \ln \frac{\rho}{\rho_0} \rho u_{/x} - a^2 \ln \frac{\rho}{\rho_0} u \rho_{/x} - a^2 \rho_{/x} u}_{= - \left( a^2 \rho \ln \frac{\rho}{\rho_0} u \right)_{/x}} \underbrace{- a^2 \rho u_{/x} - a^2 \rho_{/x} u}_{= - ( a^2 \rho u )_{/x}} = \\ & = - \left[ \left( \rho \frac{u^2}{2} + a^2 \rho \ln \frac{\rho}{\rho_0} + \rho a^2 \right) u \right]_{/x} \ . \end{aligned}$$ ``` (pde:hyperbolic:entropy:1d-system:shallow-water)= ### Shallow water (pde:hyperbolic:entropy:1d-system:euler)= ### Euler equations