The Direct Discontinuous Galerkin Method with Explicit-Implicit-Null Time Discretizations for Nonlinear Diffusion Equations

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 uₜ = (a(u)uₓ)ₓ. The basic idea of the EIN method is to add and subtract two equal constant coefficient terms a₁uₓₓ (a₁ = a₀ × maxᵤ 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 a₀ ≥ 0.5 and the third order EIN-DDG scheme is unconditionally stable under the condition a₀ ≥ 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 a₀ 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.