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 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, , that approximate the solution u at the mesh points, . The authors conduct the FDM method on , for T sufficiently large such that
where is the last point in the time mesh and is the distance between the points in the time mesh. The approximations for the steady-state solution are given by
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 , , , , where . They employ a FDM for this PDE for large times until the steady-state is reached. Why not considering the ODE, , , instead?