A novel quantum algorithm for solving advection-diffusion equation by the lattice Boltzmann method is proposed. The presented quantum algorithm is composed of two major segments. In the first segment, equilibrium distribution function, presented in the form of a non-unitary diagonal matrix, is quantum circuit implemented by using a standard-form encoding approach. For the second segment, the quantum walk procedure as a method for implementing the propagation step is applied. The constructed algorithm is presented as a series of single-and two-qubit gates, as well as multiple-input controlled-NOT gates. In order to demonstrate the validity of the proposed quantum algorithm, the unsteady one-dimensional (1D) and two-dimensional (2D) advection-diffusion equations are solved by using the IBM's quantum computing software development framework Qiskit, while the analytic solution and the classic code are used for verification. Finally, the complexity analysis and directions for future work are discussed.
A lattice Boltzmann method (LBM) is utilized to solve single-phase transient flow in pipes. In order to eliminate grid limitation related to the method of characteristics, governing equations are modified using appropriate coordinate transformation. The introduced modification removes connection between Courant number and spatial disposition of the computational nodes, forming a more flexible and robust mathematical base for numerical simulations. The computational grid is configured independently of the wave speed, significantly decreasing the demand for computational resources and maintaining the required accuracy of the method. Thereafter, the appropriate equilibrium distribution function for the D1Q3 lattice has been defined. In order to give a comprehensive base for modeling transient flow in complex pipeline systems, detailed elaboration of the corresponding boundary conditions has been given. Two benchmark problems with the corresponding error analysis are used to validate the proposed procedure.
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