We study a general convergence theory for the analysis of numerical solutions to a magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension $d$$(=2,3)$. First, we introduce the concept of dissipative weak (DW) solutions and prove the weak–strong uniqueness property for DW solutions, meaning a DW solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the DW solution, as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax equivalence theorem. Finally, as an application of this theory, we show the convergence analysis of two numerical methods.

中文翻译：

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弱无散磁场可压缩MHD系统数值解的收敛性

我们研究了一种通用收敛理论，用于分析磁流体动力学系统的数值解，该系统描述了空间维度 $d$$(=2,3)$ 中可压缩、粘性、导电流体的时间演化。首先，我们引入了耗散弱（DW）解的概念，并证明了 DW 解的弱-强唯一性，这意味着 DW 解与源自后者生命周期的相同初始数据的经典解相吻合。接下来，我们引入一致逼近的概念，并证明一致逼近对 DW 解以及经典解的收敛性。将一致近似解释为数值解的能量稳定性和一致性，我们构建了著名的 Lax 等价定理的非线性变体。最后，