  This paper develops the high-order accurate entropy stable (ES) finite
difference schemes for the shallow water magnetohydrodynamic (SWMHD)
equations.They are built on the numerical approximation of the modified SWMHD
equations with the Janhunen source term. First, the second-order accurate
well-balanced semi-discrete entropy conservative (EC) schemes are constructed,
satisfying the entropy identity for the given convex entropy function and
preserving the steady states of the lake at rest (with zero magnetic field).
The key is to match both discretizations for the fluxes and the non-flat river
bed bottom and Janhunen source terms, and to find the affordable EC fluxes of
the second-order EC schemes. Next, by using the second-order EC schemes as
building block, high-order accurate well-balanced semi-discrete EC schemes are
proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes
%satisfying the entropy inequality are derived by adding a suitable dissipation
term to the EC scheme with the WENO reconstruction of the scaled entropy
variables in order to suppress the numerical oscillations of the EC schemes.
After that, the semi-discrete schemes are integrated in time by using the
high-order strong stability preserving explicit Runge-Kutta schemes to obtain
the fully-discrete high-order well-balanced schemes. The ES property of the
Lax-Friedrichs flux is also proved and then the positivity-preserving ES
schemes are studied by using the positivity-preserving flux limiter. Finally,
extensive numerical tests are conducted to validate the accuracy, the
well-balanced, ES and positivity-preserving properties, and the ability to
capture discontinuities of our schemes.
