An Implicit Characteristic Galerkin Method for the Lattice Boltzmann Equation

To implement the lattice Boltzmann equation (LBE) method on unstructured meshes with large time steps, we present an implicit characteristic Galerkin method for the lattice Boltzmann equation (CGLBE). Unlike other approaches that directly discretize the discrete Boltzmann equation in space by finite-difference or finite-volume method, the discrete Boltzmann equation is first integrated along the characteristics to yield the general form of the lattice Boltzmann equation. Then, the standard Galerkin finite-element method is applied to the modified equation that is derived through implicit local approximations of the lattice Boltzmann equation. The Galerkin finite-element method allows easy use of the Cartesian variables on complex geometries. In addition, large time steps can be used since the resulting equation is implicit in time. Flows past a circular cylinder will be tested to estimate the accuracy and the effects of the implicit scheme. Implicit treatment of the relaxation term in the LBE will also be examined.