A high-order implicit least square-based finite difference-finite volume method for incompressible flows on unstructured grids

Abstract

In this study, a high-order implicit least squares-based finite difference-finite volume (ILSFD-FV) method with a lattice Boltzmann flux solver is presented for the simulation of two-dimensional incompressible flows on unstructured grids. In this method, a high-order polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by the least squares-based finite difference scheme. The volume integral of the high-order polynomial over the control cell results in a pre-multiplied coefficient matrix in the time-dependent term. This makes the high-order method be implicit in nature. With this feature, a high-order implicit Runge–Kutta time integration scheme, namely, the explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) scheme, is applied to obtain the time-accurate solutions for flow problems. The non-linear system of equations arising from each ESDIRK stage except for the first explicit stage is solved by a dual time stepping approach. A matrix-free lower-upper symmetric Gauss–Seidel solver is then used to efficiently march the solution in the pseudo time. The present high-order ILSFD-FV method is verified and validated by both steady and unsteady 2D incompressible flow problems. Numerical results indicate that the developed implicit method outperforms its explicit counterpart in terms of the convergence property and computational efficiency. The speedup ratio of the computational effort is about 3–22.