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

We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$ 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 ${\mathrm{\ell }}^{\mathrm{\infty }}$ 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.