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A note on methods with 0(H4) and 0(H6) phase lags for periodic
Two families of computational methods are discussed for the solution of second order periodic initial value problems.
The first is a family with 0(H4) phase-lag which contains the recently published "Numerov made explicit" method of Chawla [2]. The second is a family with 0(H6) phase-lag and periodicity interval given by H2 Є (0,12)
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Multiderivative methods for periodic initial value problems
A family of two-step multiderivative methods based on Pade approximants to the exponential function is developed. The methods are analysed and periodicity intervals in PECE mode are calculated.
Two of the methods are tested on two problems from the literature and one predictor-corrector combination is tested on two further problems
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
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