TY - JOUR T1 - The Direct Discontinuous Galerkin Method with Explicit-Implicit-Null Time Discretizations for Nonlinear Diffusion Equations AU - Li , Yumiao AU - Yang , Yin AU - Liu , Tiegang AU - Yuan , Weixiong AU - Cao , Kui JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 175 EP - 198 PY - 2025 DA - 2025/04 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0076 UR - https://global-sci.org/intro/article_detail/nmtma/23946.html KW - Direct discontinuous Galerkin method, explicit-implicit-null time discretization, stability, nonlinear diffusion equation, steady-state equation. AB -

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.