Nonconservative evolution problems describe irreversible processes and dissipative effects in a broad variety of phenomena. Such problems are often characterised by a conservative part, which can be modelled as a Hamiltonian term, and a nonconservative part, in the form of gradient flow dissipation. Traditional numerical approximations of this class of problem typically fail to retain the separation into conservative and nonconservative parts hence leading to unphysical solutions. In this work we propose a mixed variational method that gives a semi-discrete problem with the same geometric structure as the infinite-dimensional problem. As a consequence the conservation laws and the dissipative terms are retained. A priori convergence estimates on the solution are established. Numerical tests of the Korteweg-de Vries equation and of the two-dimensional Navier-Stokes equations on the torus and on the sphere are presented to corroborate the theoretical findings.

翻译：非保守演化问题描述了广泛现象中的不可逆过程与耗散效应。此类问题通常具有保守部分（可建模为哈密顿项）和非保守部分（表现为梯度流耗散形式）。针对这类问题的传统数值近似方法通常无法保持保守与非保守部分的分离，从而导致非物理解。本研究提出一种混合变分方法，该方法生成的半离散问题具有与无限维问题相同的几何结构，从而保留了守恒律与耗散项。本文建立了关于解的先验收敛性估计。通过环面和球面上的Korteweg-de Vries方程及二维Navier-Stokes方程的数值实验，验证了理论结果。