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We design an arbitrary-order free energy satisfying discontinuous Galerkin(DG) method for solving time-dependent Poisson-Nernst-Planck systems. Both thesemi-discrete and fully discrete DG methods are shown to satisfy thecorresponding discrete free energy dissipation law for positive numericalsolutions. Positivities of numerical solutions are enforced by anaccuracy-preserving limiter in reference to positive cell averages. Numericalexamples are presented to demonstrate the high resolution of the numericalalgorithm and to illustrate the proven properties of mass conservation, freeenergy dissipation, as well as the preservation of steady states.

A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson--Nernst--Planck systems