Error analysis of an Algebraic Flux Correction Scheme for a nonlinear Scalar Conservation Law Using SSP-RK2
Christos Pervolianakis
TL;DR
The paper analyzes a stabilized finite element approach for nonlinear scalar conservation laws in two dimensions, combining a lumped-mass low-order scheme with an algebraic flux correction (AFC) limiter that preserves linearity and enforces a discrete maximum principle. Temporal evolution uses the SSP-RK2 method, giving a fully discrete, linear scheme whose stability and accuracy are established under a CFL condition $k=\mathcal{O}(h^2)$. The main theoretical result provides an $L^{2}$-in-space, $\ell^{\infty}$-in-time error bound of $\|U^n-u^n\|_{L^{2}} \le C(k^2 + h^{1/2})$, with numerical experiments suggesting the spatial convergence is actually near $h^2$ in practice. The work demonstrates max-principle preservation, robustness for linear advection, and accurate nonlinear dynamics such as the inviscid Burger equation, highlighting the practical effectiveness of AFC with a linearity-preserving limiter for nonlinear hyperbolic problems.
Abstract
We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain $Ω\subset{\R}^2$ with Lipschitz boundary $\partialΩ.$ 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.