Method of Characteristics for parametric problems

Question

I am considering the equation

$u_t(t,x,\eta)+V(t,x,\eta)\cdot u_x(t,x,\eta)=0$

where $V:[0,T]\times\mathbb{R}^n\times[0,1]^D\to \mathbb{R}^n$. For the parameter independent case, one would solve this equation via the method of characteristics.

Is there also literature for the parameter-dependent case? What does the relevant ODE system look like now? Like this?

$\gamma_t(s,\eta)=V(s,\gamma(s),\eta)$

$\gamma(t_0,\eta)=x_0$

And what are the characteristic curves? Maybe

$\{(s,\eta,\gamma(s,\eta) \ | \ s \in [0,T]\}$?

Answer 1

The pde $u_t + V u_x = 0$ is a linear advection equation, where $V(t,x; \eta)$ may be dependent on time, space and a parameter $\eta$, e.g. $$ V(t,x;\eta) = e^{-\eta t} - \eta x. $$ The method of characteristics for the initial-value problem $u(0,x; \eta)=u_0(x)$ gives

• $\frac{\text d}{\text ds} t = 0$, letting $t(0)=0$ gives $t=s$;
• $\frac{\text d}{\text ds} x = V$, letting $x(0)=x_0$ gives $x$ in terms of $s$, $\eta$ and $x_0$.
• $\frac{\text d}{\text ds} u = 0$, letting $u(0)=u_0(x_0)$ gives $u=u_0(x_0)$.

The characteristic curves aren't straight lines in general, but $u$ will be constant along these curves. With the example above, we find $x(s) = (x_0 + s) e^{-\eta s}$, so that $u(t,x; \eta)=u_0(x e^{\eta t} - t)$.