Hi Rebecca!
We can approach this problem using the Rankine–Hugoniot condition, which determines whether a discontinuous function is a weak solution of a conservation law.
Step 1: Identify the given function
We are given:
u(x,y) = 3 when x - 2y > 0
u(x,y) = 1 when x - 2y < 0
The function changes abruptly across the line:
x - 2y = 0
Rearranging:
x = 2y
This is the discontinuity, or jump, line.
Step 2: Identify the propagation speed
The discontinuity travels along:
x = 2y
Therefore, its speed is:
s = dx/dy = 2
The values on either side are:
uL = 1 (left side)
uR = 3 (right side)
Step 3: Apply the Rankine–Hugoniot condition
For a conservation law of the form:
u_y + [f(u)]_x = 0
the Rankine–Hugoniot condition is:
s(uR - uL) = f(uR) - f(uL)
Substitute the values:
2(3 - 1) = f(3) - f(1)
Simplify:
4 = f(3) - f(1)
This is the key condition!
The given function is a weak solution precisely when the flux function satisfies:
f(3) - f(1) = 4
Step 4: Determine whether the function is a weak solution
For part (a), substitute the flux function from the conservation law into:
f(3) - f(1)
If the result equals 4, the Rankine–Hugoniot condition is satisfied, and the function is a weak solution.
For part (b), repeat the calculation using the second conservation law.
If the result does not equal 4, the Rankine–Hugoniot condition fails, and the function is not a weak solution.
Final Conclusion
The discontinuity travels at speed 2, and the jump in u is:
3 - 1 = 2
Therefore, the necessary and sufficient condition for this piecewise constant function to be a weak solution of a conservation law of the form u_y + [f(u)]_x = 0 is:
f(3) - f(1) = 4
To finish parts (a) and (b) numerically, we need the two conservation law equations that were omitted from the question.
I hope this helps clarify how to approach the problem!
