Error analysis of an Algebraic Flux Correction Scheme for a nonlinear Scalar Conservation Law Using SSP-RK2

Abstract

We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain $\Omega \subset {\mathbb{R}}^2$ with Lipschitz boundary $\partial \Omega .$ We discretize the spatial variable with the standard finite element method where we use a local extremum diminishing flux limiter which is linearity preserving. For temporal discretization, we use the second order explicit strong stability preserving Runge--Kutta method. It is known that the resulting fully-discrete scheme satisfies the discrete maximum principle. Under the sufficiently regularity of the weak solution and the CFL condition $k=\mathcal{O}(h^2)$, we derive error estimates in $L^2-$ norm for the algebraic flux correction scheme in space and in ${\ell }^{\infty }$ in time. We also present numerical experiments that validate that the fully-discrete scheme satisfies the temporal order of convergence of the fully-discrete scheme that we proved in the theoretical analysis.

References in corpus (1)

• An Assessment of Solvers for Algebraically Stabilized Discretizations of Convection-Diffusion-Reaction Equations