Lagged diffusivity method for the solution of nonlinear diffusion convection problems with finite differences

Abstract

This article concerns with the analysis of an iterative procedure for the solution of a nonlinear nonstationary diffusion convection equation in a two-dimensional bounded domain supplemented by Dirichlet boundary conditions. This procedure, denoted Lagged Diffusivity method, computes the solution by lagging the diffusion term. A model problem is considered and a finite difference discretization for that model is described. Furthermore, properties of the finite difference operator are proved. Then, a sufficient condition for the convergence of the Lagged Diffusivity method is given. At each stage of the iterative procedure, a linear system has to be solved and the Arithmetic Mean method is used. Numerical experiments show the efficiency, for different test functions, of the Lagged Diffusivity method combined with the Arithmetic Mean method as inner solver. Better results are obtained when the convection term increases. Statement of the problemConsider as model problem the nonlinear diffusion convection equationwhere u = u(x, y, t) is the density function at the point (x, y) at the time t of a diffusion medium R, s = s(x, y, u) > 0 is the diffusion coefficient or diffusivity and is dependent on the solution u, a = a(x, y) ≥ 0 is the absorption term,ṽ =ṽ(x, y, t, u) is the velocity vector and the source term s(x, y, t) is a real valued sufficiently smooth function.Equation (1) can be supplemented by the initial condition (t = 0)in the closureR of R and by Dirichlet boundary condition on the contour ∂R of R of the formIn the following, we suppose R to be a rectangular domain with boundary ∂R and we assume that the functions s, a, and s satisfy the "smoothness" conditions:

Advancing parabolic operators in thermodynamic MHD models II: Evaluating a Practical Time Step Limit for Unconditionally Stable Methods

“…The MAS model 1 described in Paper I (see references within) uses an overall time step set by a flow CFL condition, and operator splits several terms that would require very small time steps. These include the parabolic operators of artificial kinematic viscosity and Spitzer thermal conduction [6] given by where v, T , and ρ are the plasma flow velocity, temperature, and density respectively, ν(x) is the coefficient of viscosity, b = |B|/B is the normalized direction of the magnetic field, γ = 5/3 is the adiabatic index, m p is the proton mass, k is Boltzman's constant, κ 0 is the Spitzer coefficient for thermal conduction, f m (T ) is used to increase the parallel thermal conductivity at low temperatures which, in conjunction with an inverse modification to radiative cooling, broadens the transition region [7], f c (r) is a profile that limits the radial extent that collisional thermal conduction is active, and T 0 is the previous step's temperature used to keep the operator linear through lagged diffusivity [8].…”