A key consideration in the development of numerical schemes for time-dependent partial differential equations (PDEs) is the ability to preserve certain properties of the continuum solution, such as associated conservation laws or other geometric structures of the solution. There is a long history of the development and analysis of such structure-preserving discretisation schemes, including both proofs that standard schemes have structure-preserving properties and proposals for novel schemes that achieve both high-order accuracy and exact preservation of certain properties of the continuum differential equation. When coupled with implicit time-stepping methods, a major downside to these schemes is that their structure-preserving properties generally rely on exact solution of the (possibly nonlinear) systems of equations defining each time-step in the discrete scheme. For small systems, this is often possible (up to the accuracy of floating-point arithmetic), but it becomes impractical for the large linear systems that arise when considering typical discretisation of space-time PDEs. In this paper, we propose a modification to the standard flexible generalised minimum residual (FGMRES) iteration that enforces selected constraints on approximate numerical solutions. We demonstrate its application to both systems of conservation laws and dissipative systems.

翻译：在含时偏微分方程（PDEs）数值格式的开发中，一个关键考量是能否保留连续解的某些属性，例如相关的守恒律或解的其他几何结构。关于此类保结构离散格式的开发与分析已有悠久历史，既包括证明标准格式具有保结构性质的研究，也包括提出能够同时实现高阶精度与连续微分方程某些属性精确保持的新型格式方案。当与隐式时间步进方法结合时，这类格式的主要缺陷在于其保结构性质通常依赖于离散格式中定义每个时间步的（可能非线性的）方程组精确解。对于小规模系统，这一目标通常可行（可达浮点运算精度），但当考虑时空偏微分方程典型离散化产生的大型线性系统时，此方法变得不切实际。本文提出一种对标准柔性广义最小残差（FGMRES）迭代的改进方案，该方案可对近似数值解实施选定约束。我们通过守恒律系统和耗散系统的实例验证了该方法的有效性。