  We study a general convergence theory for the analysis of numerical solutions
to the 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 solutions and prove
the weak-strong uniqueness property for dissipative weak solutions, meaning a
dissipative weak 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 dissipative weak 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 for two numerical methods.
