System of conservation laws ($U_t+{\partial }_{x}F(U)=0$)

A system of conservation laws evolves a vector of conserved variables $U$ through a flux vector $F(U)$. For a smooth solution it has quasilinear form $U_t+A(U)U_x=0$, where $A(U)=DF(U)$ is the flux Jacobian.

Two smooth differential systems related by a nonlinear change of dependent variables need not define the same weak solutions. If $U_t+F(U{)}_x=0$, a smooth function $V=g(U)$ satisfies $V_t+g^{\prime }(U)F^{\prime }(U)U_x=0$ for smooth $U$; even when this can be written as another divergence, its Rankine-Hugoniot conditions need not be equivalent to those in $U$. The smooth chain rule does not justify multiplying jump distributions by discontinuous coefficients.

For example the shallow water equations can be written smoothly as $h_t+(hv{)}_x=0$ and $v_t+(h+v^2/2{)}_x=0$. Mechanical mass and momentum conservation instead select $h_t+(hv{)}_x=0$ and $(hv{)}_t+(h{v}^2+h^2/2{)}_x=0$. These systems agree for smooth positive $h$, but their jump laws differ. The states $(h,v)=(2,\sqrt{3}/2)$ and $(1,0)$ satisfy the mass/momentum jump laws with speed $\sqrt{3}$, while the velocity-form jump equation would demand $11/(4\sqrt{3})$. A model for discontinuities must specify the conservative variables or an equivalent admissible regularization, rather than using smooth equivalence alone.

For a strictly hyperbolic system of conservation laws $U_t+F(U{)}_x=0$ with ordered characteristic speeds ${\lambda }_1<\cdots <{\lambda }_n$, a Lax family-$j$ shock has ${\lambda }_j(U_R)<s<{\lambda }_j(U_L)$. The remaining families must stay on the same side of $s$: ${\lambda }_{j-1}(U_L)<s$ and $s<{\lambda }_{j+1}(U_R)$ where those neighbors exist. These inequalities imply that the lower families have speed below $s$ on both sides and the higher families speed above $s$ on both sides. There are $n+1$ incoming characteristic directions: two from the compressed family and one from each other family. The Rankine-Hugoniot conditions supply the jump balance, while these inequalities supply admissibility. For a convex scalar flux this reduces to the entropy shock condition.

A Riemann problem for a one-dimensional system of conservation laws ${\partial }_{t}U+{\partial }_{x}f(U)=0$ has two constant initial states, $U_L$ for $x<0$ and $U_R$ for $x>0$. The data and equation are invariant under $(x,t)\mapsto (\lambda x,\lambda t)$. When the physically admissible entropy solution is unique, this invariance makes it a self-similar solution depending on $x/t$. For ideal-gas compressible flow, the characteristic speeds are $u-c,u,u+c$. The two acoustic families carry shock waves or rarefaction waves, while the middle family carries a contact discontinuity. Sufficiently strong separating velocities can instead create a vacuum region.

Two positive uniform ideal-gas states can be joined by one nontrivial rarefaction wave precisely when their specific entropy agrees, $p_L/{\rho }_L^{\gamma }=p_R/{\rho }_R^{\gamma }$, and for one $\sigma \in \{\pm 1\}$ they satisfy $u_R-u_L=2\sigma (c_R-c_L)/(\gamma -1)$ with $\sigma (c_R-c_L)>0$. These equations are the constant-entropy and Riemann invariant conditions of the self-similar ideal-gas rarefaction. They are sufficient because its affine $u,c$ profiles match the two states at ${\xi }_L=u_L+\sigma c_L$ and ${\xi }_R=u_R+\sigma c_R$, and ${\xi }_R>{\xi }_L$. The left family has $c_R<c_L$, while the right family has $c_R>c_L$. The inequality $u_R>u_L$ alone does not impose the two matching equalities.

A contact discontinuity in one-dimensional gas dynamics moves with the common fluid velocity $s=u_L=u_R$ and has continuous pressure, while mass density and specific entropy can jump. The mass jump condition $s[\rho ]=[\rho u]$ is automatic with common $u=s$; momentum conservation $s[\rho u]=[\rho u^2+p]$ then requires $[p]=0$. The energy jump condition is also satisfied. Thus a contact is a material interface, not a compression shock. A smooth self-similar positive-density flow cannot spread this zero-speed-relative-to-the-fluid family into a fan: $u=x/t$ on an interval would require both $du/d(x/t)=1$ and, from continuity, $du/d(x/t)=0$.

A conservation law flux $F(U)$ transports the conserved density $U$ across a surface. In one dimension $U_t+{\partial }_{x}F(U)=0$; a moving discontinuity satisfies $s[U]=[F]$. For a stationary shock wave, each conserved flux has the same value on both sides.

A first-order quasilinear system has the form $U_t+A(U,x,t)U_x=S(U,x,t)$: derivatives of the unknown enter linearly, while their coefficient matrix may depend on the unknown itself.

A quasilinear system is hyperbolic where its principal coefficient matrix has real eigenvalues and enough independent eigenvectors. The eigenvalues are its characteristic speeds.

The flux Jacobian is the Jacobian matrix of the flux in a system of conservation laws. Its eigenvalues are the local characteristic speeds, and its left and right eigenvectors select the corresponding wave variables and directions in state space.

A characteristic speed ${\lambda }_j(U)$ is an eigenvalue of the flux Jacobian $A(U)=DF(U)$ for a system of conservation laws. In a classical solution of $U_t+A(U)U_x=0$, let ${\ell }_j(U)$ be a left eigenvector. Along $\dot{x}={\lambda }_j(U(t,x))$, the chain rule givesFor a constant matrix, $w_j={\ell }_j\cdot U$ therefore obeys $(w_j{)}_t+{\lambda }_j(w_j{)}_x=0$. For example, $A=c\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ gives $w_{+}=U_1+U_2$ moving at $c$ and $w_{-}=U_1-U_2$ moving at $-c$.

For a nonlinear system, ${\ell }_j(U)\cdot dU$ need not be the differential of a scalar function, even after multiplication by a nonzero scalar factor. A scalar Riemann invariant requires such an additional integrability property. Consequently the projected identity alone does not supply a globally defined transported scalar for every wave family.

A kinematic wave is a disturbance whose propagation speed is determined by the local relation between a conserved density and its flux. In a scalar conservation law $u_t+f(u{)}_x=0$, smooth values travel at speed $f^{\prime }(u)$.

Let $H(t)={\int }_0^{t}E(r)dr$, increasing until $t^{\ast }$ and decreasing afterwards. For dry initial data and zero upstream depth, a characteristic emitted at time $\tau$ has $h=H(t)-H(\tau )$ and position $s=\frac{5}{3}{\int }_{\tau }^t[H(r)-H(\tau ){]}^{2/3}dr$, for as long as its depth stays nonnegative. The height follows by integrating the source along a characteristic, and the position follows from the derivative of the Manning flux.

When $E<0$ and $H(t)>0$, define the earlier emission time ${\tau }_d(t)\le t^{\ast }$ by $H({\tau }_d)=H(t)$. The front where this water is just exhausted hasIt moves downstream because $s_d^{\prime }=-(10/9)E(t){\int }_{{\tau }_d}^t[H(r)-H(t){]}^{-1/3}dr>0$. Behind it depth is zero; ahead, unexhausted characteristic labels determine the wet profile. This construction presumes removal limited by water availability: continuing a negative source unchanged at zero depth would violate nonnegativity. If $H$ reaches zero, the remaining initially generated water has also vanished.

With constant lateral supply $M$, a discharge law of the displayed form gives equilibrium area $[(m+1)(Q_{\mathrm{in}}+Ms)/c{]}^{1/(m+1)}$ and characteristic speed $c{A}^m$. Under a Chezy law, a wetted-perimeter relation $P_b\propto A^q$ gives $m=(1-q)/2$. Small area perturbations are transported downstream with amplitude inversely proportional to this characteristic speed; finite compressive disturbances can still form shocks because the flux is convex.