solving partial differential equations remarks in and problemswhere the cpu time to compute solution of pde can be hours and daysit is very important to utilize symmetry as we do above to reduce the size of the problem also note the remarks in exercise about the constant area under the xtcurveherethe area is and : as : (if the mesh is sufficiently fine one will get convergence to smaller values for small if the mesh is not fine enough to properly resolve thin-shaped initial conditionexercise compute solutions as many diffusion problems reach stationary time-independent solution as the model problem from sect is one example where xtd td const for when does not depend on timethe diffusion equation reduces to vu xd /in one dimensionand vr /in and this is the famous poisson equationor if it is known as the laplace equation in this limit there is no need for an initial conditionbut the boundary conditions are the same as for the diffusion equation we now consider one-dimensional problem xd / cu ld ( which is known as two-point boundary value problem this is nothing but the stationary limit of the diffusion problem in sect how can we solve such stationary problem ( )the simplest strategywhen we already have solver for the corresponding time-dependent problemis to use that solver and simulate until which in practice means that xtno longer changes in time (within some tolerancea nice feature of implicit methods like the backward euler scheme is that one can take one very long time step to "infinityand produce the solution of ( alet ( be valid at mesh points xi in spacediscretize by finite differenceand set up system of equations for the point values ui , where ui is the approximation at mesh point xi bshow that if in ( ( )it leads to the same equations as in acdemonstrateby running programthat you can take one large time step with the backward euler scheme and compute the solution of ( the solution is very boring since it is constantu xd filenamerod_stationary py remarks if the interest is in the stationary limit of diffusion equationone can either solve the associated laplace or poisson equation directlyor use backward euler scheme for the time-dependent diffusion equation with very long time step using forward euler scheme with small time steps is typically inappropriate in