Analytical Solutions as Benchmarks
Exact closed-form solutions of simplified problems provide ground truth for verification without needing a manufactured source.
Ground Truth from Mathematics
A handful of physics problems have exact closed-form solutions: steady conduction in simple geometries, linear waves, potential flow, diffusion of a point source. These solutions are ground truth. Running a code on such a problem and comparing to the exact answer verifies the code on a physically meaningful case, with no manufactured source term needed.
Advantages over Manufactured Solutions
- The test case is physical, so it exercises the code in a regime it will actually be used in.
- No artificial source term is added, so the test checks the equations as written.
- Many analytic solutions have known asymptotics that reveal how the code behaves near boundaries and singularities.
Limits
Analytic solutions exist only for simplified problems, usually linear, low-dimensional, and with simple geometry. They cannot exercise the nonlinear coupling and complex geometry that a design calculation actually uses. That is precisely why manufactured solutions complement them: manufactured solutions reach where analytic ones cannot, at the cost of adding an artificial source. A strong suite uses both.
Using Them Well
The comparison should be quantitative and mesh-resolved: not merely that the code looks right, but that it converges to the analytic solution at the expected order. An analytic benchmark used only for a single-mesh eyeball check wastes most of its value. Run on a refinement sequence, it doubles as an order-of-accuracy test with a physical solution.
Special-function solutions, such as those built from Bessel functions or error functions, extend the reach of analytic benchmarks to cylindrical and diffusion problems relevant to reactor and plasma geometries. Their exact values are available to high precision, so they make excellent references. The discipline is the same as for any verification: compare with tolerance, refine the mesh, and confirm the rate, not just the value.