Algorithm for a cost-reducing time-integration scheme for solving incompressible Navier–Stokes equations

In this paper, we propose a novel trajectory-approximation technique as a time-integration scheme in a semi-Lagrangian framework, which is generally applicable to solve advectional partial differential equations in engineering and physics. The proposed trajectory-approximation technique resolves strong nonlinearity in the Cauchy problem and saves computational costs in comparison with the existing third-order methods by reducing the number of interpolations occurring at every spatial lattice point for each time step. Moreover, an explicit formula is introduced as a more efficient form of the proposed time-integration scheme. To obtain numerical evidence, we apply the proposed method to simulate four benchmark test flows of incompressible Navier–Stokes equations: a linear advection–diffusion, a flow on a square domain, a shear layer flow, and a backward-facing step flow. The proposed method provides third-order accuracy in terms of both time and space in the overall backward semi-Lagrangian methodology. It also demonstrates superior performance over recently developed third-order trajectory-approximation schemes in terms of the efficiency and execution time in solving the Cauchy problem with strong nonlinearity.