TY - JOUR
T1 - Fourth-Order Splitting Methods for Time-Dependant Differential Equations
AU - Jürgen Geiser
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 321
EP - 339
PY - 2008
DA - 2008/01
SN - 1
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/nmtma/6054.html
KW - Partial differential equations, operator-splitting methods, evolution equations, ADI
methods, LOD methods, stability analysis, higher-order methods.
AB - <p style="text-align: justify;">This study was suggested by previous work on the simulation of
evolution equations with scale-dependent processes, e.g.,
wave-propagation or heat-transfer, that are modeled by wave
equations or heat equations. Here, we study both parabolic and
hyperbolic equations. We focus on ADI (alternating direction
implicit) methods and LOD (locally one-dimensional) methods, which
are standard splitting methods of lower order, e.g. second-order.
Our aim is to develop higher-order ADI methods, which are performed
by Richardson extrapolation, Crank-Nicolson methods and higher-order
LOD methods, based on locally higher-order methods. We discuss the
new theoretical results of the stability and consistency of the ADI
methods. The main idea is to apply a higher-order time
discretization and combine it with the ADI methods. We also discuss
the discretization and splitting methods for first-order and
second-order evolution equations. The stability analysis is given
for the ADI method for first-order time derivatives and for the LOD
(locally one-dimensional) methods for second-order time derivatives.
The higher-order methods are unconditionally stable. Some numerical
experiments verify our results.</p>