# Guidelines for choosing manufactured solutions for numerical PDE schemes

When testing a numerical method for a PDE, I know that it's often useful to compare it to a known analytical solution. If none is available, one can always 'manufacture' a solution, substitute it into the PDE and obtain the source term, boundary conditions, and initial conditions needed for the problem.

But what guidelines should I follow for choosing an appropriate 'manufactured' solution to test the scheme on? What qualities should the manufactured solution have?

After some thought, I've come up with the following considerations:

1. Sufficient Smoothness - Whatever the highest order derivative is in the PDE (for any variable), the manufactured solution should have at least this degree of smoothness within the domain of the problem.
2. Physically Plausible Coefficients - Non-negative values for coefficients that are physically positive.
3. Appropriate Scaling - A non-dimensional version of the problem should be used, if possible.

However, my main concern is manufacturing a solution that can adequately capture the theoretical order of accuracy. Is there anything else i should consider in the properties of the manufactured solution to ensure that the theoretical order of accuracy will be captured?

Are there anything else I should generally consider when choosing a manufactured solution?

• (1) could be rephrased as requesting a classical solution. It's usually preferred to choose smooth ($C^\infty$) manufactured solutions so that (if implemented correctly) you can expect to attain the design order of accuracy for any convergent discretization. – Jed Brown Jun 15 '14 at 4:29