When a finite difference scheme is applied to a PDE, it does not solve the original equation exactly — it solves a different PDE entirely, known as the modified equation. This equation accounts for the truncation error introduced by the discretisation and reveals the true physics being simulated. Odd-order derivative terms correspond to advection and dispersion; even-order terms correspond to diffusion. The coefficients of the modified equation allow one to make rational arguments about stability, numerical diffusion, and numerical dispersion.

For a linear PDE with constant coefficients and a corresponding difference scheme, the modified equation takes the general form

where the coefficients \(a_i\) are not, in general, equal to the coefficients of the original PDE. The leading even-order coefficient \(a_2\) governs numerical diffusion; the leading odd-order coefficient beyond \(a_1\) governs numerical dispersion.

In two spatial dimensions the modified equation takes the form

Note the cross-derivative term \(a_{11}u_{xy}\), which is absent from the 1D analysis. It introduces anisotropic diffusion — the scheme behaves differently in different spatial directions. The coefficients \(a_{jl}\) carry a double index \((j,l)\) denoting the order of differentiation in \(x\) and \(y\) respectively.