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Numerical schemes provably preserving the positivity of density and pressureare highly desirable for MHD, but the rigorous positivity-preserving (PP)analysis remains challenging. The difficulties mainly arise from the intrinsiccomplexity of the MHD equations as well as the indeterminate relation betweenthe PP property and the divergence-free condition on magnetic field. We presentthe first rigorous PP analysis of conservative schemes with Lax-Friedrichs (LF)flux for ideal MHD. The significant innovation is the discovery of theoreticalconnection between PP property and a discrete divergence-free (DDF) condition.This connection is established through the generalized LF splitting properties,which are alternatives of the usually-expected LF splitting property that doesnot hold for ideal MHD. The generalized LF splitting properties involve anumber of admissible states strongly coupled by DDF condition, making theirderivation very difficult. We derive these properties via a novel equivalentform of the admissible state set and an important inequality skillfullyconstructed by technical estimates. Rigorous PP analysis is presented forfinite volume and discontinuous Galerkin schemes with LF flux on uniformCartesian meshes. In 1D case, PP property is proved for the first-order schemewith proper numerical viscosity, and also for arbitrarily high-order schemesunder conditions accessible by a PP limiter. In 2D case, we show that the DDFcondition is necessary and crucial for achieving PP property. It is observedthat even slightly violating the proposed DDF condition may cause failure topreserve the positivity of pressure. We prove that the 2D LF type scheme withproper numerical viscosity preserves the positivity and DDF condition.Sufficient conditions are derived for 2D PP high-order schemes, and extensionto 3D is discussed. Numerical examples confirm the theoretical findings.

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics