# Numerical method for steady-state solution to viscous Burgers' equation

I am reading a paper in which a specific partial differential equation (PDE) on the space-time domain $$[-1,1]\times[0,\infty)$$ is studied. The authors are interested in the steady-state solution. They design a finite difference method (FDM) for the PDE. As usual, there are certain discretizations in time-space, $$U_j^n$$, that approximate the solution $$u$$ at the mesh points, $$u(x_j,t_n)$$. The authors conduct the FDM method on $$[-1,1]\times [0,T]$$, for $$T$$ sufficiently large such that $$\left|\frac{U_j^N-U_j^{N-1}}{\Delta t}\right|<10^{-12},\quad \forall j,$$ where $$t_N=T$$ is the last point in the time mesh and $$\Delta t$$ is the distance between the points in the time mesh. The approximations for the steady-state solution are given by $$\{U_j^N\}_j$$.

I wonder why the authors rely on the PDE to study the steady-state solution. As far as I know, the steady-state solution comes from equating the derivatives with respect to time to $$0$$ in the PDE. The remaining equation is thus an ordinary differential equation (ODE) in space. To approximate the steady-state solution, one just needs to design a FDM for this ODE, which is easier than dealing with the PDE for sure. Is there anything I am not understanding properly?

For completeness, I am referring to the paper Supersensitivity due to uncertain boundary conditions. The authors deal with the PDE $$u_t+uu_x=\nu u_{xx}$$, $$x\in (-1,1)$$, $$u( -1,t)=1+\delta$$, $$u(1,t)=-1$$, where $$\nu,\delta>0$$. They employ a FDM for this PDE for large times until the steady-state is reached. Why not considering the ODE $$uu'=\nu u''$$, $$u(-1)=1+\delta$$, $$u(1)=-1$$, instead?

You are right. Indeed, solving the ODE for steady-state (time-independent) solutions $$(\tfrac12 u^2)' = \nu u''$$ is rather straightforward, and we have $$\tfrac12 u^2 = \nu u' + k_1 \, ,$$ where $$k_1$$ is an integration constant. Thus, integrating the previous ODE gives $$u(x) = \sqrt{2 k_1} \tanh \left( \sqrt{\frac{k_1}{2}} \frac{k_2 - x}{\nu} \right) ,$$ where $$k_2$$ is an integration constant. The two integration constants are obtained by imposing the boundary conditions, and the sensitivity of the solution with respect to $$\delta$$ can be investigated.