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Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. (English) Zbl 1113.65095

The topic of the paper is a fairly novel methodology for treating transport equations, in this case the nonlinear Dirac equation for the spinor field. The paper starts by proving some conservation properties of the exact (continuous) equation, and then introduces the discontinuous Galerkin approximation. Here an entropy inequality for the semi-discrete solution is shown. The time-discretisation is achieved with a 3rd order total variation diminishing Runge-Kutta scheme, and the working of everything is demonstrated in some numerical experiments of ternary collisions, observing some interesting phenomena.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35L60 First-order nonlinear hyperbolic equations
35Q40 PDEs in connection with quantum mechanics
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