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.

翻译：我们考虑定义在有界区域$\Omega\subset{\R}^2$上的一个具有线性和非线性通量函数的标量守恒律，其边界$\partial\Omega$为Lipschitz边界。空间变量采用标准有限元方法进行离散，其中使用了一个保持线性性的局部极值递减通量限制器。对于时间离散，我们采用二阶显式强稳定保持Runge--Kutta方法。已知由此得到的全离散格式满足离散最大值原理。在弱解具有足够正则性且满足CFL条件$k = \mathcal{O}(h^2)$的前提下，我们推导了空间上采用代数通量修正格式、时间上采用$\ell^\infty$范数的$L^{2}-$范数误差估计。我们还给出了数值实验，验证了全离散格式满足我们在理论分析中所证明的时间收敛阶。