A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation
Résumé
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 $L^2$ norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of $c$, the speed of light. Moreover, when $c$ 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 $c \DX \to 0$). 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.
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