  We develop in this paper two classes of length preserving schemes for the
Landau-Lifshitz equation based on two different Lagrange multiplier approaches.
In the first approach, the Lagrange multiplier $\lambda(\bx,t)$ equals to
$|\nabla m(\bx,t)|^2$ at the continuous level, while in the second approach,
the Lagrange multiplier $\lambda(\bx,t)$ is introduced to enforce the length
constraint at the discrete level and is identically zero at the continuous
level. By using a predictor-corrector approach, we construct efficient and
robust length preserving higher-order schemes for the Landau-Lifshitz equation,
with the computational cost dominated by the predictor step which is simply a
semi-implicit scheme. Furthermore, by introducing another space-independent
Lagrange multiplier, we construct energy dissipative, in addition to length
preserving, schemes for the Landau-Lifshitz equation, at the expense of solving
one nonlinear algebraic equation. We present ample numerical experiments to
validate the stability and accuracy for the proposed schemes, and also provide
a performance comparison with some existing schemes.
