TY - JOUR
T1 - Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations
AU - Zhu , Liyong
JO - Advances in Applied Mathematics and Mechanics
VL - 1
SP - 157
EP - 172
PY - 2018
DA - 2018/05
SN - 9
DO - http://doi.org/10.4208/aamm.2015.m1045
UR - https://global-sci.org/intro/article_detail/aamm/12142.html
KW - Exponential Runge-Kutta method, explicit scheme, linear splitting, discrete fast Fourier transforms, Allen-Cahn equation.
AB - <p style="text-align: justify;">In this paper we develop explicit fast exponential Runge-Kutta methods for
the numerical solutions of a class of parabolic equations. By incorporating the linear
splitting technique into the explicit exponential Runge-Kutta schemes, we are able to
greatly improve the numerical stability. The proposed numerical methods could be
fast implemented through use of decompositions of compact spatial difference operators
on a regular mesh together with discrete fast Fourier transform techniques. The
exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations
with variable time step sizes, as well as applied to stiff nonlinearity and
boundary conditions of different types. Linear stabilities of the proposed schemes and
their comparison with other schemes are presented. We also numerically demonstrate
accuracy, stability and robustness of the proposed method through some typical model
problems.</p>