A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the (Formula presented.) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of (Formula presented.), the speed of light. Moreover, when (Formula presented.) diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a "diffusive limit" emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that (Formula presented.)). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed. © 2014 Springer Science+Business Media Dordrecht.