  The Landau--Lifshitz--Baryakhtar equation describes the evolution of magnetic
spin field in magnetic materials at elevated temperature below the Curie
temperature, when long-range interactions and longitudinal dynamics are taken
into account. We propose two linear fully-discrete $C^1$-conforming methods to
solve the problem, namely a semi-implicit Euler method and a semi-implicit BDF
method, and show that these schemes are unconditionally stable. Error analysis
is performed which shows optimal convergence rates in each case. Numerical
results corroborate our theoretical results.

Numerical Analysis;Numerical Analysis