Abstract
We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
Highlights
The Fokker–Planck equation (FPE), known as Smoluchowski equation or Kolmogorov forward equation, is one of the most important equations in theoretical physics and applied mathematics with application in physical chemistry, protein synthesis, plasma physics, semiconductor device simulation and others
In the context of finite volume (FV) methods, the central objective is a robust and accurate discretization of the flux implied by the FPE
A important discretization scheme for the flux was derived by Scharfetter and Gummel [47] and independently by Allan and Southwell [1] and Il’in [29]
Summary
The Fokker–Planck equation (FPE), known as Smoluchowski equation or Kolmogorov forward equation, is one of the most important equations in theoretical physics and applied mathematics with application in physical chemistry, protein synthesis, plasma physics, semiconductor device simulation and others. An alternative flux discretization method, called square-root approximation (SQRA) scheme, has been derived explicitly for high dimensional problems in molecular dynamics [33] It was independently obtained from a maximum entropy path principle [13] and from discretizing the Jordan–Kinderlehrer–Otto variational formulation of the FPE [40]. The Bernoulli function B0,−1 interpolates between the appropriate discretizations for the drift- and diffusion-dominated limits, which is why the SG scheme is the preferred FV scheme for Fokker–Planck type operators. Convergence of order O(h) for general B-schemes including SG, SQRA as well as Stolarsky means has been proved in 1D [35]. Convergence for the SQRA discretization has been investigated in [40] in 1D, Donati et al [14] (formally, rectangular meshes) and [26] using G-convergence on grids with random weights
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