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:
- 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.
- Physically Plausible Coefficients - Non-negative values for coefficients that are physically positive.
- 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?