Burgers' Equation

BurgersEquation{D, T} implements the inviscid Burgers' equation in $D$ dimensions:

\[\frac{\partial u}{\partial t} + \sum_{d=1}^D \frac{\partial}{\partial x_d}\left(\frac{u^2}{2}\right) = 0 .\]

The state is a scalar (M = 1). Information travels with velocity $(u, \dots, u)$, so compressive regions steepen into shocks and expansive regions open into rarefaction fans.

In a configuration: :PDE_name => :burgers, :PDE_D => D.

Implementation

• Variables. prim2cons and cons2prim are the identity.
• Flux. $F_d(u) = u^2/2$ in every direction.
• Wave speed. max_eigenvalue returns $|u|$; velocity returns the $1 \times 1$ Jacobian $u$.
• Moving grids. fluid_velocity returns $(u, \dots, u)$, the characteristic velocity, so a Lagrangian grid follows the characteristics.

Exact solutions

Riemann and smoothed Riemann (any dimension)

Along the normal coordinate $\xi = (x - p_0)\cdot n$ the equation reduces to a 1D law with flux $a\,u^2/2$, where $a = \sum_d n_d$. The characteristic speeds are $a u_L$ and $a u_R$:

• $a u_L > a u_R$: a shock moving at $a (u_L + u_R)/2$ (Rankine–Hugoniot).
• otherwise: a rarefaction fan with $u = \xi / (a t)$ between the two speeds.

The wave type depends on the sign of $a$, so flipping the normal and swapping the states gives the same solution. The normal does not need to be normalized. For SRiemann the closure returns the sharp Riemann solution, a good reference once the waves are wider than width.

Gauss (any dimension)

The solution is given implicitly by $u = u_0(x - u\,t\,\mathbf{1})$, which is solved with Newton's method. This is only valid before the pulse breaks, at

\[t^* = \frac{\mathrm{width}}{|a|}\sqrt{\frac{e}{2D}} .\]

Past $t^*$ Newton may not converge; the closure then warns and returns its last iterate.

Box (1D)

The top-hat case (u_box > u_bg) produces a rarefaction at the left edge and a shock at the right edge. The well case (u_box < u_bg) produces a shock at the left edge and a rarefaction at the right edge. Both are exact before and after the time the fan catches up with the shock, $t_\mathrm{int} = 2(x_e - x_s)/|u_\mathrm{box} - u_\mathrm{bg}|$. After that the shock moves like $\sqrt{t}$ relative to the background flow and its strength decays like $t^{-1/2}$.

Other initial conditions

There is no exact solution. The closure warns and returns the initial condition, which is only correct at $t = 0$.

API

BurgersEquation{D, T} <: HyperbolicPDE{D, 1, T}