Semidiscretization

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interior grid values satisfyingBoundary values complete this ordinary differential equation system; time has not yet been replaced by a sequence of steps.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]