A matrix-free, implicit finite volume lattice Boltzmann method for steady flows

- A matrix-free, implicit finite volume lattice Boltzmann method is proposed;
- A hybrid of the LUSGS and the Jacobi iteration is proposed to solve the system;
- The storage of the Jacobian matrixes in the present scheme is completely eliminated.

In the present paper, a matrix-free, implicit finite volume lattice Boltzmann method for steady flow on unstructured mesh is proposed. The approximate linear system arising from the implicit finite volume discretization of lattice Boltzmann equation (LBE) is solved by a novel algorithm, which combines the lower-upper symmetric Gauss-Seidel (LU-SGS) and the Jacobi iteration schemes. A remarkable feature of the present implicit method is that the storage of the Jacobian matrixes of the convection and collision terms can be completely eliminated by approximating the Jacobian matrix-solution incremental vector product with appropriate numerical flux incremental vector and the numerical increment of collision term, resulting a matrix-free implicit scheme. The present method is validated by several two-dimensional testing cases.