Exponential Time Integrator for Solving the Lattice Boltzmann Equations Based on a Spectral-Element Discontinuous Galerkin Approach

I'll present a high-order time integration method for solving lattice Boltzmann equation (LBE). We use high-order spectral-element discretization in space based on discontinuous Galerkin approach and apply a Krylov subspace approximation for time-advancing. The semi-discrete form of the spectral-element discontinuous Galerkin (SEDG) method on the LBE brings us to an ODE of the form $\raise0.7ex\hbox{${\partial U}$} \!\mathord{\left/ {\vphantom {{\partial U} {\partial t}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\partial t}$}\,\,=\,\,-AU$ with initial condition$U(0)=U_0 $ where $U$is a solution vector. A is a large sparse matrix based on a polynomial approximation order N. The solution of the equation is $U(t)=U_0 e^{-At}$. The explicit one-step method is based on the computation of matrix functions of the type $U(t+\delta )=U(t)e^{-A\delta }$. We project the matrix exponential and the solution vector onto a finite dimensional Krylov subspace $K_m $ of order $m$. We use the Arnoldi algorithm to generate an orthogonal basis $V_m $ and an Hessenberg matrix$H_m $ for approximating $e^{-A\delta }U(t)\,\,\approx \,V_m e^{-H_m \delta }V_m^T U(t)$. We will study convergence of the exponential time integrator for possible use of larger time step with high-order $m. $We will demonstrate its efficiency and accuracy compared to the Runge-Kutta time-stepping methods.