Are there any one dimensional hyperbolic test cases for a 1D uniform flow?

(I need the test case for the purpose of showing the satisfaction of geometric conservation law on a moving mesh).


1 Answer 1


P.S. The answer is edited after first remark below it.

If one means by "1D uniform flow" a passive transport where the velocity is constant, i.e. you want to solve the advection equation $u_t + c u_x = 0$, then the test cases typically differ only by different choices of initial conditions, say the function $u(x,0)$, e.g. a constant impulse (i.e. $u$ equals zero everywhere expect on a finite interval where it is equal to 1), a Gaussian (e.g. $u(x,0)=e^{-x^2}$), or a piecewise linear function (e.g. a triangle shape). You can test the geometric property that the measure of support of solution is preserved in time (e.g. the interval where $u(x,t)$ is positive is simply shifted by speed $c$), see e.g. CLAWPACK.

If one does not want to solve the advection equation $u_t +c u_x = 0$, but some "geometric transformation", e.g. to move a mesh, then the constant speed of flow means that any subset of the initial mesh is simply shifted to a new position by the shift $c t$, where $t$ is time.

If one means by uniform flow in fact the steady flow, then in the 1D case of advection equation it means to solve $u_t+c(x) u_x = 0$, then I have tested one example with $c(x)=2+\sin(x)$. The exact solution for a given initial function has quite complicated form, it can be found sometimes by e.g. Mathematica, but one property can be checked easily for any initial function - the exact solution is periodic in the sense $u(x,\pi/\sqrt3)=u(x-2 \pi,0)$. I include an animated gif, where $u(x,0)=\sin(x)$ for $x \in (0,4 \pi)$ and the animation runs for $t \in (0,\pi/\sqrt3)$ with periodic boundary conditions:

enter image description here

You might check in such simulations that the position of intervals where e.g. $u(x,t)$ is positive is shifted from its initial position at $t=0$ by factor $2 \pi$ at $t=\pi/\sqrt3$, similarly for intervals where $u(x,0)$ is negative.

If you compute directly a moving mesh with this speed $c(x)=\sin(x) + 2$, then any subset of initial mesh shall be shifted to the right by factor $2 \pi$ after one time period $t=\pi/\sqrt3$.


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