By means of a modified equation analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error.
Lemma 3.3 below deals with the higher-order estimates of the solutions which are needed to guarantee the extension of a local classical solution to a global one.
As the existence of the higher-order derivatives of the solution is not supposed, we need to use the difference quotient for the rigorous derivation of the higher-order estimates.
By using Lemma 1.1, we find that δ n = π + O ( 1 n 11 ) and ω n = π + O ( 1 n 9 ), which provide the higher-order estimates for the constant π.
Higher-order estimates of the density, velocity and magnetic field can be obtained in a standard way provided that (|mathrm {u}|_{H^{1}}) and (|mathrm {H}|_{H^{1}}) are uniformly bounded with respect to time.
In the next lemma, we give some high-order estimates.
It has been demonstrated that the method is capable of delivering higher-order estimates of the eigenvalues at a very low cost; see [24].
