Spatial Discretization and Boundaries

Spatial discretization replaces a differential operator and its domain by a finite numerical problem. The replacement must preserve the relevant inner product, boundary physics, and continuum limit. This chapter develops those choices for Schrödinger, Dirac, and Klein–Gordon computations, including outgoing waves and nonuniform grids. Eigensolvers and time integrators act on the resulting representation in the spectral and propagation chapters.

Enter this chapter

Begin with boundary conditions in computation and finite-difference Schrödinger operators. A particle in a box provides a transparent first experiment: the physical answer is known, the discrete spectrum can be obtained analytically, and grid refinement can be distinguished from eigensolver accuracy.

The representation and reliability chapter supplies array conventions, dimensionless variables, residuals, and convergence studies. Keep a declaration of the wavefunction measure beside the discretization. A matrix that looks nonsymmetric may be self-adjoint in a weighted metric; a matrix that is exactly Hermitian may still approximate the wrong physical boundary condition.

Choose a spatial approximation

• Finite differences give sparse local stencils and explicit dispersion errors. Matrix Numerov develops a compact higher-order construction and explains the distinction between a symmetric kinetic matrix and an apparently generalized matrix pencil.
• Shooting turns a one-dimensional boundary-value problem into root finding. Matching joins independently propagated solutions and avoids some exponentially ill-conditioned long integrations. Neither method is made reliable solely by a small root-finder tolerance.
• Finite elements build a weak-form stiffness and overlap problem from local basis functions. They are useful when geometry, local refinement, or a variational formulation matters.
• Spectral and pseudospectral methods use global Fourier or polynomial functions. Discrete variable representations use compatible quadrature to connect basis coefficients with weighted node values. Smoothness, aliasing, and potential quadrature determine whether the expected accuracy is attained.
• Coordinate transformations redistribute resolution while tracking Jacobians, kinetic operators, and endpoint domains. They complement the preceding methods rather than replacing their convergence checks.

For escaping packets, absorbing boundaries explains transparent conditions, masks, and probability accounting. The complex absorbing potential page adds reflection calibration and resonance trajectories; the scattering chapter handles amplitudes and outgoing spectral problems.

Relativistic grids need additional checks

Dirac equation numerics treats spinor ordering, weighted Hermiticity, energy sectors, and propagation. Klein–Gordon numerics treats the second-order evolution, frequency branches, charge, and stability bound. Their continuum interpretations belong in Relativistic Quantum Mechanics.

Lattice discretization develops dispersion and gauge-link checks. Fermion doubling explains why a norm-conserving centered Dirac difference can introduce extra low-energy species and how a Wilson term changes the grid spectrum. A stable update does not establish the correct particle content.

A comparison that distinguishes the methods

Use $H=-\partial_x^2$ on $0<x<1$ with Dirichlet endpoints. The exact ground energy is $\pi^2$. At mesh spacing $h$, the standard centered difference gives

$$E_h^{\mathrm{FD}}=\frac4{h^2}\sin^2\frac{\pi h}{2}<\pi^2.$$

A conforming linear finite-element calculation with exact consistent mass gives

$$E_h^{\mathrm{FE}}= \frac{6[1-\cos(\pi h)]}{h^2[2+\cos(\pi h)]}>\pi^2.$$

Both converge quadratically for this smooth problem, from opposite sides. The finite-element upper bound follows from its variational construction; it is not a property of every Hermitian discretization. Derivations and boundary qualifications appear on the individual method pages.

A useful comparison records grid or basis size, boundary locations, matrix residual, normalization defect, and error relative to the analytic value. Extend the comparison to a smooth potential and then a nonsmooth one. A change in convergence rate is information about the approximation, not a reason to force every data series into the same power law.

References

• Brenner, Susanne C., and L. Ridgway Scott. The Mathematical Theory of Finite Element Methods. 3rd ed. Springer, 2008.
• LeVeque, Randall J. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007.
• Trefethen, Lloyd N. Spectral Methods in MATLAB. SIAM, 2000.