Discretization Error
The gap between the exact solution of the equations and the solution on a finite mesh, usually the dominant numerical error.
The Cost of Being Finite
Continuous equations are solved on finite meshes and finite time steps, and the discrete solution differs from the exact continuous one. That difference is the discretization error. For most well-behaved calculations it is the largest of the numerical errors, larger than round-off and iterative error, so estimating and controlling it is the heart of solution verification.
How It Scales
For a scheme of formal order p, the discretization error scales as the mesh spacing raised to the power p. A second-order scheme has error proportional to spacing squared, so halving the spacing cuts the error by four. This scaling is what convergence and order-of-accuracy studies measure, and it is what makes the error predictable enough to extrapolate away.
Estimating It
- Run the problem on at least three systematically refined meshes.
- Confirm the observed order matches the formal order.
- Use Richardson extrapolation to estimate the exact solution and hence the error on each mesh.
When Scaling Breaks
The clean power-law scaling assumes a smooth solution and a mesh in the asymptotic range, fine enough that the leading error term dominates. Discontinuities, singularities, and under-resolved features break the assumption, reducing the observed order and invalidating simple extrapolation. Sharp gradients near a boundary or a shock are common culprits. Detecting a broken scaling is itself useful: it tells you the mesh is too coarse to trust, regardless of how good the answer looks.
Discretization error is the line item that verification most directly controls, and the one that most often contaminates a naive comparison to experiment. Bounding it first is what lets a validation discrepancy be attributed to physics rather than to an under-resolved mesh.