Riemann Problem for viscous burgers equation

Question

I am trying to find a weak form or analytic solution to compare numerical methods with for the viscous Burgers' Equations $$ u_t -\nu u_{xx}+uu_x = 0$$ subject to initial conditions $u(0,t)=u(1,t)=0$ and $$ u(x,0)=\begin{cases} 1 \hspace{4mm} x\in (0,\frac{1}{2}]\\ 0 \hspace{4mm}x\in(\frac{1}{2},1) \end{cases}$$ I see that weak form solutions exist for Burgers' equation $u_t + uu_x= 0$ with the same initial conditions given here: Prove that shock wave is weak solution of Burgers' equation (Riemann problem). Could I just use that the solution in the previous question satisfiess $u_{xx}=0$ or is it just not that simple?

Answer 1

It's not that simple. You have diffusion, which means that shock waves do not occur. Diffusion has the effect of smoothing things out. It might give you some insight if you solve the pure heat equation $u_t=\nu u_{xx}$ with the same initial data first.