Finite-dimensional truncations are routinely used to approximate partial differential equations (PDEs), either to obtain numerical solutions or to derive reduced-order models. The resulting discretized equations are known to violate certain physical properties of the system. In particular, first integrals of the PDE may not remain invariant after discretization. Here, we use the method of reduced-order nonlinear solutions (RONS) to ensure that the conserved quantities of the PDE survive its finite-dimensional truncation. In particular, we develop two methods: Galerkin RONS and finite volume RONS. Galerkin RONS ensures the conservation of first integrals in Galerkin-type truncations, whether used for direct numerical simulations or reduced-order modeling. Similarly, finite volume RONS conserves any number of first integrals of the system, including its total energy, after finite volume discretization. Both methods are applicable to general time-dependent PDEs and can be easily incorporated in existing Galerkin-type or finite volume code. We demonstrate the efficacy of our methods on two examples: direct numerical simulations of the shallow water equation and a reduced-order model of the nonlinear Schrodinger equation. As a byproduct, we also generalize RONS to phenomena described by a system of PDEs.

翻译：有限维截断通常用于近似偏微分方程（PDE），无论是为了获取数值解还是推导降阶模型。已知由此得到的离散化方程会违反系统的某些物理性质，特别是PDE的一阶积分可能在离散化后不再保持不变。本文采用降阶非线性解方法（RONS）确保PDE的守恒量在其有限维截断后得以保留。我们具体发展了两种方法：伽辽金RONS和有限体积RONS。伽辽金RONS可确保伽辽金型截断（无论是用于直接数值模拟还是降阶建模）中一阶积分的守恒性；类似地，有限体积RONS能在有限体积离散化后保持系统任意数量的一阶积分（包括总能量）。两种方法均适用于一般时变PDE，并可便捷地集成到现有伽辽金型或有限体积代码中。我们通过浅水方程的直接数值模拟和非线性薛定谔方程的降阶模型这两个示例验证了方法的有效性。作为副产品，我们还将其推广至由偏微分方程组描述的物理现象。