@Article{NMTMA-18-175,
author = {Li , YumiaoYang , YinLiu , TiegangYuan , Weixiong and Cao , Kui},
title = {The Direct Discontinuous Galerkin Method with Explicit-Implicit-Null Time Discretizations for Nonlinear Diffusion Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2025},
volume = {18},
number = {1},
pages = {175--198},
abstract = {<p style="text-align: justify;">This paper proposes a discussion of the direct discontinuous Galerkin
(DDG) methods coupled with explicit-implicit-null time discretizations (EIN) for
solving the nonlinear diffusion equation $u_t = (a(u)u_x)_x.$ The basic idea of the EIN
method is to add and subtract two equal constant coefficient terms $a_1u_{xx}$ $(a_1 =
a_0 ×{\rm max}_u a(u))$ on the right-hand side of the above equation, and then apply the
explicit-implicit time-marching method to the equivalent equation. The EIN method
does not require any nonlinear iterative solver while eliminating the severe time-step restrictions typically associated with explicit methods. We present the stability
criterion of the EIN-DDG schemes for the simplified equation with periodic boundary
conditions via the Fourier method, where the first order and second order EIN-DDG
schemes are unconditionally stable when $a_0 ≥ 0.5$ and the third order EIN-DDG
scheme is unconditionally stable under the condition $a_0 ≥ 0.54.$ Numerical experiments show the stability and optimal orders of accuracy of our proposed schemes
with a relaxed time-step restriction and the appropriate coefficient $a_0$ for both linear and nonlinear equations in one-dimensional and two-dimensional settings. Furthermore, we also show that the computational efficiency of our EIN-DDG schemes
and explicit Runge-Kutta DDG (EX-RK-DDG) schemes for steady-state equations with
small viscosity coefficients to illustrate the effectiveness of the present methods.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2024-0076},
url = {http://global-sci.org/intro/article_detail/nmtma/23946.html}
}