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Structure-preserving discretization of the Poisson-Nernst-Planck equations via the Onsager principle
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DOI：10.1016/j.jcp.2026.114877.png)
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Abstract
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We propose a variational approach for designing energy-stable and positivity-preserving numerical schemes for the Poisson-Nernst-Planck (PNP) equations. By leveraging the Onsager variational principle, the governing equations are reformulated as a constrained minimization problem. A straightforward time discretization naturally yields a mass-transport formulation and ensures mass conservation and energy dissipation. This structure is preserved under spatial discretization, which transforms the problem into a well-posed minimization problem with linear constraints. We rigorously prove that the scheme guarantees positivity, mass conservation and unconditional energy stability, even in high-dimensional settings. Theoretical results are supported by numerical tests which demonstrate the accuracy, stability, and robustness of the proposed method.
Keywords:
Onsager variational principle
PNP systems
Optimization
Dynamic transport
Positivity
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3.8
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1.5W
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7.4W
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