Error analysis of an Algebraic Flux Correction Scheme for a nonlinear Scalar Conservation Law Using SSP-RK2
<|reference_start|>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 $\Omega\subset{\R}^2$ 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.<|reference_end|>
@article{pervolianakis2024error,
    title={Error analysis of an Algebraic Flux Correction Scheme for a nonlinear
  Scalar Conservation Law Using SSP-RK2},
    author={Christos Pervolianakis},
    journal={arXiv preprint arXiv:2409.18606},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.18606},
    primaryClass={math.NA cs.NA}
}