Positivity 2: 193–219, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Two-Step Fifth-Order Methods for Evolutionary
Problems with Positive Operators
ANDREI I. TOLSTYKH
Computing Center of Russian Academy of Sciences, Moscow Vavilova str. 40, Russia
(Received: 9 July 1997; Accepted: 4 February 1998)
Abstract. The family of three-level ﬁfth-order time integrators are considered for hyperbolic, par-
abolic PDEs and stiff ODEs. They are classiﬁed into two parts depending on positivity or negativity of
the operators corresponding to the second time derivatives. Two options are presented for hyperbolic
equations. Stability analysis is performed for linear cases. It is shown that the scheme for ODEs
can be nearly A-stable and nearly L-stable for particular values of its free parameter. Numerical
illustration is presented for hyperbolic case.
Mathematics Subject Classiﬁcation (1991): 65M06, 65M12, 65L05, 65L06, 65L20
Key words: ﬁfth-order schemes, hyperbolic and parabolic PDEs, stiff ODEs
When constructing difference schemes for convection or convection-diffusion prob-
lems, it was found to be proﬁtable to use in many practical cases the high-order
non-centered Pade-type (“compact”) differencing  as approximations to spatial
derivatives. Using the computational ﬂuid dynamics terminology, the technique
was named Compact Upwind Differencing (CUD). This type of differencing re-
sults in positive operators which provide negligible phase and amplitude errors of
numerical solutions in wide domains of wave numbers admitted by meshes as well
as a natural ﬁlter of high frequency numerical noise.
When applied to steady-state problems, CUD-based methods show high accu-
racy and the capability of good resolution of ﬁne details of high-gradients solutions
even if relatively coarse meshes are used.
However, to use CUD (or other high-order) approximations to spatial deriva-
tives in the case of unsteady problems, high-order time stepping techniques may
be needed if one wants to preserve high accuracy of time-dependent numerical
solutions. To be more speciﬁc, we are interested in time integrators which can
provide the accuracy comparable with that of the ﬁfth-order CUD approximations
. A possible approach here is to use ﬁfth-order six-stage Runge-Kutta methods.
However, we prefer to construct its alternative which does not require formulations
of the boundary conditions for intermediate stages.
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