This work provides reliable a posteriori error estimates for Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems. The classes of systems we study are quite general with a focus on convection-dominated and degenerate parabolic problems. Our a posteriori error bounds are valid for a family of discontinuous Galerkin spatial discretizations and various temporal discretizations that include explicit and implicit-explicit time-stepping schemes, popular tools for practical simulations of this class of problem. We prove that our estimators provide reliable upper bounds for the error of the numerical method and present numerical evidence showing that they achieve the same order of convergence as the error. Since one of our main interests is the convection dominant case, we also track the dependence of the estimator on the viscosity coefficient.

翻译：本研究为非线性对流-扩散系统的龙格-库塔间断伽辽金近似提供了可靠的后验误差估计。所研究的系统类具有相当的普遍性，重点关注对流占优和退化抛物问题。我们的后验误差界适用于一类间断伽辽金空间离散化及多种时间离散格式，包括显式和隐显式时间步进方案——这类问题实际模拟中常用的工具。我们证明了所提出的估计子能为数值方法的误差提供可靠上界，并通过数值实验表明其收敛阶与误差阶相同。由于对流占优情形是我们的主要关注点之一，我们还追踪了估计子对粘性系数的依赖关系。