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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]

We present a hydrodynamic model for nonisothermal, incompressible binary fluids that incorporates buoyancy effects using the Boussinesq approximation. This model adheres to the generalized Onsager principle, ensuring both volume preservation for each fluid phase and a positive entropy production rate under thermodynamically consistent boundary conditions, maintaining overall thermodynamic consistency. We then develop a set of second-order numerical algorithms that preserve both volume and entropy production rates to solve the model in finite domains, subject to physically appropriate boundary conditions. By implementing an efficient adaptive time-stepping strategy, we perform several numerical simulations that validate the accuracy and second-order convergence of the schemes. These simulations successfully capture Rayleigh-Bénard convection in binary fluids and the interfacial dynamics between immiscible fluids under the influence of temperature gradients, buoyancy, and interfacial forces, demonstrating the accuracy and practical utility of the model and the numerical methods. [ABSTRACT FROM AUTHOR]