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We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can also overcome the non-physical issues that arise from many traditional numerical schemes. Due to their preservation of physical nature, these schemes can be straightforwardly used as backbones for further developing randomized and quantum algorithms in simulating Lindblad equations. In this work, we systematically study this family of structure-preserving deterministic schemes and perform a detailed error analysis, which is validated through numerical examples. [ABSTRACT FROM AUTHOR]