scholarly journals Use of Proper Closure Equations in Finite Volume Discretization Schemes

10.5772/38692 ◽  
2012 ◽  
Author(s):  
A. Ashrafizadeh ◽  
M. Rezvani ◽  
B. Alini
Keyword(s):  
Finite Volume ◽  
Finite Volume Discretization ◽  
Discretization Schemes

scholarly journals Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator

2021 ◽  
Author(s):  
Martin Heida ◽  
Markus Kantner ◽  
Artur Stephan
Keyword(s):  
Finite Volume ◽  
Numerical Experiments ◽  
Point Of View ◽  
Weight Functions ◽  
Convergence Properties ◽  
Finite Volume Discretization ◽  
Discretization Schemes ◽  
Stolarsky Means ◽  
Fokker Planck

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.

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