The method of lines and semidiscrete systems

For the heat equation on[0,1] with constant diffusivity kappa>0 and homogeneous Dirichlet boundaries, take equally spaced nodes x_j=jh with h=1/m. Replace the second derivative at each interior node by the centered three-point difference. The unknowns U_j(t) remain functions of continuous time, and the boundary values are fixed at zero. Nonzero prescribed boundaries would enter as forcing terms; treating them as free interior unknowns changes the problem and its energy balance.

The resulting ordinary differential equation is U′=kappa L_h U. For smooth exact solutions, inserting nodal samples into this system leaves a spatial residual of order h squared, provided the fourth spatial derivative is uniformly bounded on the time interval. This is a consistency statement, not yet a global solution-error estimate. A time integrator introduces another error source, whose constants may depend on the grid through the stiffness of L_h. A tolerance or time step appropriate on one grid need not remain appropriate after spatial refinement.

Define the grid L2 norm squared as h times the sum of squared interior values. Multiplying each semidiscrete equation by hU_j and summing performs a discrete integration by parts. Boundary terms vanish because the endpoint values are zero, leaving minus kappa over h times the sum of squared adjacent differences. Thus the semidiscrete energy cannot increase. The norm includes h so that it approximates a spatial integral as the grid is refined, rather than artificially growing with the number of degrees of freedom.

The centered Dirichlet Laplacian has sine eigenvectors. The k-th eigenvalue is negative four over h squared times the squared sine of k pi h divided by two. Low fixed modes approximate continuum decay rates, while the largest-magnitude eigenvalues scale like h to the power minus two. This separation makes the system increasingly stiff as h decreases. The energy proof controls the exact semidiscrete flow in the grid norm; it does not authorize arbitrary steps of an explicit ODE method.

Forward Euler multiplies an eigenmode by one plus time step times its semidiscrete eigenvalue. Stability requires that multiplier to have magnitude at most one for every mode, not just for the smooth initial profile one happens to test. A sufficient heat restriction is kappa times time step divided by h squared at most one half. On a particular finite Dirichlet grid the exact endpoint of the spectrum can permit a slightly larger step, but the familiar restriction is uniform and also gives nonnegative stencil weights.

Implicit methods can remove that explicit stability restriction, but they still need time resolution for accuracy and sufficiently accurate linear solves. Error analysis should identify spatial and temporal norms, boundary assumptions, regularity and the interval of comparison. The worked example deliberately compares a low-mode decay rate with a high-mode explicit instability: a solver may appear accurate on smooth data while hidden high-frequency perturbations are amplified by its time discretization.