The SQRA Operator: Convergence Behaviour and Applications

The Squareroot Approximation Operator (SQRA) is a numerical FV-operator that has recently been derived by M. Weber and coworkers and has the form of a discrete spatial chemical master equation. We use methods from stochastic homogenization to prove convergence in the context of high dimensional numerical implementation. We furthermore show that the SQRA is equivalent with the Scharfetter-Gummel scheme and use this insight to prove convergence of order 1 of both schemes in low dimensional settings. This is particularly possible due to a deep connection between the SQRA and the gradient structure of the Fokker-Planck equation discussed by Jordan, Kinderlehrer and Otto. We finally discuss physical implications of our insights and possible future applications to hydrodynamic limits arising in the modelling of organic semiconductors.