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

TL;DR

This paper analyzes the error of an algebraic flux correction scheme combined with SSP-RK2 for nonlinear scalar conservation laws, providing theoretical error estimates and validating them through numerical experiments.

Contribution

It offers the first error estimates in $L^2$ and $ ext{l}^ ext{infinity}$ norms for this flux correction scheme with SSP-RK2, under certain regularity and CFL conditions.

Findings

The scheme satisfies the discrete maximum principle.

Error estimates are derived under CFL condition $k=O(h^2)$.

Numerical experiments confirm the theoretical convergence order.

Abstract

We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain $\Omega\subset{\R}^2$ 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.