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A finite difference method approximates a differential equation by replacing derivatives with weighted combinations of nearby sampled function values. Finite difference coefficients are chosen by matching low-degree polynomials or a Taylor expansion. The resulting unknowns live on a grid, where the differential equation becomes a system of algebraic update or boundary equations.

For the one-dimensional heat equation $u_t=\nu u_{xx}$ with $\nu >0$, a forward Euler method in time and a second-order central difference in space giveFor $0\le r\le 1/2$, the coefficients are nonnegative and sum to one, so each new value is a convex combination of neighboring values. This yields a maximum norm contraction for compatible homogeneous boundaries.^[ref]

Consistency of a numerical method measures how accurately the grid equation reproduces a smooth exact solution; stability of a numerical method controls propagation of perturbations. Both matter for numerical convergence. A correct interior formula also needs a compatible boundary closure of a difference scheme and suitable initial conditions.

Hans Petter Langtangen coauthored a text on finite difference methods for partial differential equations with Svein Linge. Its treatment of diffusion equations derives discrete updates and investigates their stability. The connection between an explicit Euler method and a spatial second-difference operator makes the restriction on the time step visible in a reproducible computational model.^[ref]

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interior grid values satisfyingBoundary values complete this ordinary differential equation system; time has not yet been replaced by a sequence of steps.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

Svein Linge coauthored a text on finite difference methods with Hans Petter Langtangen. The book links mathematical discretization with implementations for partial differential equations, including diffusion equations. Deriving an update formula alongside its stability conditions explains why a physically smoothing equation can nevertheless produce an unstable numerical calculation if the time step is too large.^[ref]