 Manufactured solution

A manufactured solution is a deliberately chosen smooth function used to construct a differential equation problem with a known exact answer. Substituting it into the differential equation determines the forcing, while its boundary and initial values determine matching data. The resulting computation tests a numerical implementation against that answer.

For example, on $0<x<1$, choose $u_{\ast }(x,t)=e^{-t}\sin (\pi x)$ and consider the forced heat equationUse zero Dirichlet boundary conditions and initial data $u(x,0)=\sin (\pi x)$. Direct differentiation verifies that $u_{\ast }$ solves this problem. At a fixed final time, compute an error norm against $u_{\ast }$ on successively refined grids. A second-order spatial scheme should show a factor near four when the spatial step is halved, once time, solver and roundoff errors are smaller than the spatial error. This checks the implemented equations and their order of a numerical method; it does not establish that those equations model a particular physical system correctly.