We study a general convergence theory for the numerical solutions of compressible viscous and electrically conducting fluids with a focus on numerical schemes that preserve the divergence free property of magnetic field exactly. Our strategy utilizes the recent concepts of dissipative weak solutions and consistent approximations. First, we show the dissipative weak--strong uniqueness principle, meaning a dissipative weak solution coincides with a classical solution as long as they emanate from the same initial data. Next, we show the convergence of consistent approximation towards the dissipative weak solution and thus the classical solution. Upon interpreting the consistent approximation as the stability and consistency of suitable numerical solutions we have established a generalized Lax equivalence theory: convergence $\Longleftrightarrow$ stability and consistency. Further, to illustrate the application of this theory, we propose two novel mixed finite volume-finite element methods with exact divergence-free magnetic field. Finally, by showing solutions of these two schemes are consistent approximations, we conclude their convergence towards the dissipative weak solution and the classical solution.
翻译:我们研究的是压缩粘稠液和电流的数值解决方案的一般趋同理论,重点是保持磁场无差异特性的数字方案。我们的战略利用了消散性弱溶液和一致近似的最新概念。首先,我们展示了消散性弱强独特性原则,意思是消散性弱溶液与传统溶液相吻合,只要它们来自相同的初始数据。接着,我们展示了一致接近消散性弱溶液和经典溶液的趋同性。在将一致近似解释为我们所建立的普遍拉克斯等值理论的稳定性和一致性:趋同以美元/美元/美元/美元/美元/美元/次偏偏向型的稳定性和一致性。此外,为了说明这一理论的应用,我们提议了两种新型的混合有限量定要素方法,与完全无差异的磁场相匹配。最后,通过展示这两种办法的解决方法是一致的近似,我们得出它们与消散性弱溶液和经典溶液的趋同性。