Discretization

Discretization replaces a continuous mathematical problem by equations for finitely many unknowns. A spatial grid samples a function; a finite element method instead chooses coefficients in a finite-dimensional trial space. A time discretization supplies update equations at selected times. These choices specify what is computed, before any roundoff error is introduced.

For the heat equation $u_t=u_{xx}$, a uniform spatial grid $x_j=jh$ givesThis is a semidiscretization: time remains continuous. Applying the Forward Euler method gives a fully discrete updateFor a smooth exact solution, the normalized local truncation error is $O(k+h^2)$. Convergence additionally requires stability of a numerical method; for this one-dimensional update, $0\le k/h^2\le 1/2$ makes each new value a convex combination of neighboring values. Spatial and temporal refinement therefore interact, rather than giving two independent accuracy knobs.

Semidiscretization discretizes some variables of a differential equation while leaving others continuous. A spatial discretization of the heat equation $u_t=\kappa u_{xx}$ gives interior grid values satisfyingBoundary values complete this ordinary differential equation system; time has not yet been replaced by a sequence of steps.

Applying explicit Euler later gives a fully discrete method with stability restriction $\kappa \Delta t/\Delta x^2\le 1/2$ for the standard uniform-grid diffusion model. Keeping the two stages separate makes spatial approximation error distinct from time-integration error and allows a different time solver to be used with the same spatial operator.^[ref]