We give a priori error estimates of second order in time fully explicit Runge-Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions $\tau\leq c h$ for piecewise linear and $\tau\lesssim h^{4/3}$ for higher order finite elements, we prove a convergence rate for the energy norm $\|\cdot\|_{L^\infty_tL^2_x}+|\cdot|_{L^2_tH^{\lambda/2}_x}$ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.

翻译：针对一维空间中标量分数阶守恒律的光滑解，本文给出了使用迎风通量的二阶时间全显式龙格-库塔间断伽辽金格式的先验误差估计。在时间步长限制条件$\tau\leq c h$（针对分片线性元）和$\tau\lesssim h^{4/3}$（针对高阶有限元）下，我们证明了能量范数$\|\cdot\|_{L^\infty_tL^2_x}+|\cdot|_{L^2_tH^{\lambda/2}_x}$的收敛速率，该速率对于足够光滑的解和通量函数而言是最优的。我们的证明依赖于精确解的一种新型迎风投影。