Accuracy analysis of Lattice Boltzmann Method for advection-diffusion equation in two dimensions

I analyze the accuracy of two-dimensional lattice Boltzmann method for advectiondiffusion equation. The Chapman-Enskog expansion indicates that the lattice Boltzmann method has an error term in the diffusion process. To eliminate the error term, I introduce a parameter q which allows to control the propagation rate between neighboring sites. The advection and diffusion behavior of the lattice Boltzmann method is investigated by solving 2-D benchmark problem of a Gaussian hill in a uniform velocity field. The numerical solutions show that the schemes are indeed consistent with advection diffusion, and the error term performs as numerical diffusion. The von Neumann stability analysis method shows that the stability region of the present method shrinks by the influence of the elimination of the extra term

Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L 2 -energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples

An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is $O(N^{-r}+\tau^2)$, where $N, \tau, r$ are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids

A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime

The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the P\'eclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the approximation order, the problem's stability and perturbations in data. The convergence order is the same for both norms, but the $H^1$-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.Comment: 21 pages, 6 figures; in v2 we added two remarks and an appendix on psiDOs, and made some minor correction