discretization error

Discretization error is the truncation error you incur specifically by replacing a continuous problem — defined on a continuum of points, with derivatives and integrals — by a discrete one defined on a finite grid of points, with sums and differences. It is the gap between the answer to the continuous problem and the answer to the discrete problem you actually hand to the computer.

Continuous mathematics lives on infinitely many points; a computer can only hold finitely many. So you lay down a mesh with spacing h (in space) or a step size (in time), approximate each derivative by a finite-difference formula, each integral by a quadrature sum, and the differential equation by an algebraic system at the grid points. Each of these swaps introduces error that scales with the spacing — typically O(h^2) for a second-order scheme — so the discrete solution converges to the true continuous one as the grid is refined. For instance, approximating the heat equation's second derivative by (u(x-h) - 2 u(x) + u(x+h))/h^2 carries an O(h^2) discretization error at every grid point.

Discretization error matters because it is usually the dominant, controllable error in solving differential and integral equations, and because refining the grid to shrink it costs work — halving h in 3D can multiply the unknowns eightfold. A central practice, the order-of-accuracy study, is to refine the grid and confirm the error falls at the rate the method promises; if it does not, your code or scheme has a bug. Note the contrast with round-off: refining the grid reduces discretization error but increases total arithmetic, so round-off slowly accumulates — refinement helps only up to a point.