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Abstract

This research addresses the numerical simulation of the Boltzmann transport equation for semiconductor devices by proposing a multidimensional self-adaptive numerical simulation framework. This framework is applied to two important generalized forms of the equation: a parabolic equation with singular properties on the unit disk and a continuity equation. The study enhances the alignment of numerical simulations with physical characteristics through polar coordinate transformation and variable drift-diffusion coefficients. Innovatively, a multidimensional adaptive mesh partitioning strategy for radius-angle-time is designed and combined with an adjustable finite difference scheme to construct a highly adaptive numerical simulation method. In the construction of discrete schemes, the Swartztrauber-Sweet method and the control volume method are employed to effectively eliminate the origin singularity caused by polar coordinate transformation. On the programming front, a parallelized MATLAB algorithm is developed to optimize code execution efficiency. Numerical comparative experiments demonstrate that the adaptive method improves the accuracy of the parabolic equation by 1 to 7 times and that of the continuity equation by 10% to 70% while maintaining computational efficiency, significantly enhancing numerical simulation accuracy with high stability. Furthermore, this study systematically verifies the algorithm's convergence, stability, and parameter sensitivity using error visualization and other means. It also explores optimal parameters and establishes tuning optimization criteria. The research provides theoretical support for high-precision and highly adaptive methods in semiconductor device simulation, demonstrating outstanding advantages in handling singular regions.