We address the numerical treatment of source terms in algebraic flux correction schemes for steady convection-diffusion-reaction (CDR) equations. The proposed algorithm constrains a continuous piecewise-linear finite element approximation using a monolithic convex limiting (MCL) strategy. Failure to discretize the convective derivatives and source terms in a compatible manner produces spurious ripples, e.g., in regions where the coefficients of the continuous problem are constant and the exact solution is linear. We cure this deficiency by incorporating source term components into the fluxes and intermediate states of the MCL procedure. The design of our new limiter is motivated by the desire to preserve simple steady-state equilibria exactly, as in well-balanced schemes for the shallow water equations. The results of our numerical experiments for two-dimensional CDR problems illustrate potential benefits of well-balanced flux limiting in the scalar case.

翻译：针对稳态对流-扩散-反应（CDR）方程代数通量修正格式中的源项数值处理问题，本文提出一种基于整体凸限制（MCL）策略的连续分段线性有限元逼近约束算法。若对流导数与源项采用非相容方式离散，将在连续问题系数恒定且精确解呈线性的区域诱发伪振荡。为消除此缺陷，我们将源项分量纳入MCL流程中的通量与中间态构造过程。新限制器的设计理念源于对稳态平衡态精确保持的追求，类似于浅水方程平衡格式中采用的策略。二维CDR问题的数值实验结果表明，标量情形下的平衡通量限制技术可显著提升计算性能。