Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems

We study three multistep implicit–explicit (IMEX) time discretizations coupled with a local discontinuous Galerkin (LDG) spatial discretization for a class of one-dimensional nonlinear fourth-order partial differential equations (PDEs), with particular emphasis on the Kuramoto–Sivashinsky (KS) equation. The nonlinear convection is treated explicitly, whereas the linear second-, third-, and fourth-derivative terms are advanced implicitly. Under the quasi-uniform mesh assumption and a uniform boundedness assumption on the exact and numerical solutions, energy estimates yield L2 stability for the three schemes under a time-step restriction τ≤τ0, where τ0 is independent of the spatial mesh size. For all three schemes, a coupled LDG elliptic projection and energy estimates give the optimal spatial rate O(hN+1) together with temporal rates O(τ), O(τ2), and O(τ3), provided the multistep starting values have the corresponding accuracy. The error analysis uses a coupled LDG elliptic projection so that the discrete auxiliary error equations remain homogeneous; the residual lower-order projection terms, including the β-term, are estimated explicitly. An independent periodic manufactured-solution test verifies temporal rates approaching one, two, and three. A separate periodic spatial-refinement test at T=0.5 with the third-order formula recovers the expected O(hN+1) spatial behavior. For a long-time periodic KS test, kinetic-energy histories and spatial spectra are compared under both time-step and mesh refinement. The latter diagnostics are interpreted as robustness checks over a finite observation window rather than as evidence of a statistically stationary chaotic state or pointwise long-time convergence.