On convergence of numerical solutions for the compressible MHD system

Abstract: 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. First, we introduce the concept of consistent approximation mimicking the density positivity, energy stability, and consistency of a suitable numerical approximation. Further, we introduce the concept of dissipative weak solution, which can be obtained as the weak limit of the consistent approximation. Here, by “dissipative" we mean that the energy inequality contains energy defects that control the oscillations in the momentum equation. Next, by using the relative energy functional, we prove the dissipative weak-strong uniqueness principle, meaning that a dissipative weak solution coincides with a classical solution of the same problem as long as the latter exists. This indicates that a consistent approximation converges unconditionally to the classical solution. As a summary, we built a nonlinear variant of the Lax equivalence theory for the compressible MHD system. Finally, to show the application of the convergence theory, we propose two numerical methods. We show that the numerical solutions preserve the positivity of density and stability of the total energy. Then by using the a priori estimates derived from the energy estimates we prove that the numerical methods are consistent. Consequently, our numerical methods belong to the class of consistent approximation. Applying the prebuilt convergence theory, we conclude that the solutions of our numerical methods converge to i) the dissipative weak solution; ii) the classical solution as long as the classical solution exists. As a byproduct of the first convergence, we prove the global in-time existence of the dissipative weak solution.

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor).