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Singularity-free Numerical Scheme for the Stationary Wigner Equation

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Abstract

For the stationary Wigner equation with inflow boundary conditions, its numerical convergence with respect to the velocity mesh size are deteriorated due to the singularity at velocity zero. In this paper, using the fact that the solution of the stationary Wigner equation is subject to an algebraic constraint, we prove that the Wigner equation can be written into a form with a bounded operator $\mathcal{B}[V]$, which is equivalent to the operator $\mathcal{A}[V]=\Theta[V]/v$ in the original Wigner equation under some conditions. Then the discrete operators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the mesh size. Based on the therectical findings, a signularity-free numerical method is proposed. Numerical reuslts are proivded to show our improved numerical scheme performs much better in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.

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... Furthermore, it is found and discussed in [31] that singularity exists around v = 0 and it may bring difficulties to solve the SWIBVP numerically. In [31], the authors propose an integral constraint along with the Wigner equation to remove the singularity at v = 0, and thus a regularized SWIBVP is considered. In this paper, the integral constraint is transformed into a penalty term appending to Φ[θ ] and a modified optimization model is thus generated. ...
... It is found in literatures (e.g. [14,31]) that the SWIBVP has singularity at v = 0, which causes dilemmas when solving the SWIBVP numerically since the Wigner function will tend to infinity when v → 0. Based on the results shown in [31], an additional equation ...
... It is found in literatures (e.g. [14,31]) that the SWIBVP has singularity at v = 0, which causes dilemmas when solving the SWIBVP numerically since the Wigner function will tend to infinity when v → 0. Based on the results shown in [31], an additional equation ...
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Partial Differential Equations
  • L C Evans
L.C. Evans. Partial Differential Equations. American Mathematical Society, Providence RI, 2nd edition, 2010.